Material Balance Relations

If represents the molar flow rate of component i in the vapor phase, L = L r, the total vapor flow rate, a the interfacial area per unit volume of froth. Ay the froth height, and Afj the active bubbling area, then the component material balance for the vapor phase may be written as 

In view of Eq. 12.1.2, the right members on both sides of Eq. 12.1.3 cancel each other and we get 

This result is valid even when 0, that is, when we have nonequimolar transfer. 

Equations 12.1.4 must be integrated over the froth height to yield the composition 

The material balance relations presented above are valid for any number of components. We shall discuss solutions to this system of equations for binary mixtures in the remainder of this section of Chapter 12 before moving on to obtain generalized results for multicomponent systems in Section 12.2. In the analyses that follow we shall ignore the effects of heat transfer between the vapor and liquid phases. 

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The concentration of sodium ions is the same as the initial concentration of the NaOH, [NaOH]initial, and so the material-balance relation is [Na4] = [NaOH inma. It follows that... 

For semibatch operation, the term fraction conversion is somewhat ambiguous for many of the cases of interest. If reactant is present initially in the reactor and is added or removed in feed and effluent streams, the question arises as to the proper basis for the definition of /. In such cases it is best to work either in terms of the weight fraction of a particular component present in the fluid of interest or in terms of concentrations when constant density systems are under consideration. In terms of the symbols shown in Figure 8.20 the fundamental material balance relation becomes ... 

If the initial condition of the reactor contents is known and if the feedstream conditions are specified, it is possible to solve equation 8.6.1 to determine the effluent composition as a function of time. The solution may require the use of material balance relations for other species or a total material balance. This is particularly true of variable volume situations where the following overall material balance equation is often useful. 

The equation describing the steady-state material balance for tubular packed bed reactors can be obtained from the more general relation (12.7.28) by omitting the terms corresponding toTadial transport of matter. Hence the material balance relation becomes... 

Depending on the operational constraints, one of the two equations 12.7.48 or 12.7.49 or the choice of isothermal behavior must be used, together with the general material balance relation (equation 12.7.39), to determine the composition and temperature profiles along the length of the reactor. 

The tracer should be stable and conserved, so that it can be accounted for by a material balance relating the response to the input if the tracer decays (e.g., a radio-labeled tracer), its half-life should be such that tm (tracer) > 251 (fluid). 

For the tracer material-balance relation, equation 19.3-3 is written as... 

Material balances relating concentration and temperature of adiabatic reaction in a CSTR are obtained for several different rate equations or conditions. The curves are drawn with feed concentration Caf = 1 and residence time t = 1 in the equation,... 

The information flow diagram (Fig. 3.27) for this system shows the two component material balance relations to be linked by the equilibrium and transfer rate relationships. 

Material balances can be written for moieties which are conserved during the reaction, such as the atoms of a particular element or the total charge, or for reactant or product species if the stoichiometry is unambiguous. Oxidation-reduction reactions may be particularly troublesome. In the following situation, for example, one cannot write a material balance relating protons to water molecules. Consider the oxidation of O2 to H2O and the equilibrium dissociation of I O. 

The fouling process may be characterized as a rate dependent kinetic process describable by the material balance relation... 

A material balance relates the different measured quantities of the output curve of a step input... 

The yield of HNO2 from water decomposition, G (HN02), can be calculated by using the yields of water decomposition products and their material balance relation ... 

The second equation takes note of material balance, a principle based on the fact that all the added solute must be accounted for even though it is now present as ions. In this example, material balance means that the concentration of Cl- ions is equal to the concentration of the HC1 added initially (all the HC1 molecules are deprotonated). If we denote the numerical value of that initial concentration by [HCl]initial, the material balance relation is [Cl-] = [HCl]initia. We can combine it with the preceding equation to write... 

The material balance relation (Eq. 5.4.3) is used to eliminate the mole fractions (x ) to give t/(xo)... 

For a binary system the material balance relation (Eq. 12.1.6) may be written for component 1 as... 

The above equations are combined with the following equations to complete the VLLE model An overall material balance relates the feed flow rate F to the liquid rates and the vapor rate V ... 

The phase equilibrium and material balance relations are applied to stage 2 using Equations 3.6 and 3.7 ... 

If the equilibrium ceases to be unfavorable, with r < 1, dx/dZ will take on a negative slope. As this is prohibited by the material-balance relation, the breakthrough curve for equilibrium must be drawn continuously vertical in the concentration region for which r < 1. 

In a recycle cascade such as Fig. 12.13, feed, product, and tails quantities and compositions (the external variables) must satisfy the material-balance relations... 

Two material-balance relations among internal variables may also be written for each stage. Consider the portion of the cascade from the product end down to, but not including, stage i ... 

Terms in yp—ZF F w canceled out because of the material-balance relations... 

The internal streams between this pair of stages, Afj i and IV,-, may be expressed in terms of the streams external to this pair of stages M,-, lV,-+i, X/c, M,- 2, and Nj—i by means of the material-balance relations ... 

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