Mass balances integral

Since the reactor is assumed to be uniform in composition everywhere, we can make an integral mass balance on the number of moles Nj of species j in a reactor of volume V. This gives... 

This will be the most used forni of the mass-balance equation in the CSTR in this book. Students should either memorize this equation or preferably be able to derive it from an integral mass balance on the reactor. This equation requires three major assumptions ... 

If the liquid as well as the gas phases are ideally mixed (i. e. the reactor behaves like a completely mixed STR, often called CSTR) the degradation processes can easily be calculated from integral mass balances for ozone and the dissolved target substances,... 

The mass transfer coefficient can be found using the mass halance given by equation 3-17, simplified for QL = 0, and, if ozone decay is negligible, rL = 0. The integrated mass balance is then ... 

In deriving the integral mass balance for a closed system in Section 4.2c we eliminated the input and output terms, since by definition no mass crosses the boundaries of a closed system. It is possible, however, for energy to be transferred across the boundaries as heat or work, so that the right side of Equation 7.3-1 may not be eliminated automatically. As with mass balances, however, the accumulation term equals the final value of the balanced quantity (in this case, the system energy) minus the initial value of this quantity. Equation 7,3-1 may therefore be written... 

A very effective way to evaluate the equilibrium is to calculate the overall mass balances by numerical integration. Area A in Fig. 6.18 is equivalent to the solute accumulated inside the plant and the column, which is split into the liquid and the adsorbent phase. The integral mass balance allows the calculation of the loading, qu = q(c"), if the status I and the total porosity are known ... 

This value is in agreement with the one derived from band profiles calculated with the equilibrium-dispersive model [9]. The time given by Eq. 16.20 provides useful information regarding the specifications for the experimental conditions under which staircase binary frontal analysis must be carried out to give correct results in the determination of competitive isotherms. The concentration of the intermediate plateau is needed to calculate the integral mass balances of the two components, a critical step in the application of the method (Chapter 4). This does not apply to single-pulse frontal analysis in which series of wide rectangular pulses are injected into the column which is washed of solute between successive pulses. 

This chapter deals essentially with the apphcations of the theory of chromatography to the calculation of solutions of the SMB model in different cases of general interest. The theoretical tools required are a general model of the SMB process and a model for its columns. The former is an integral mass balance that is easy to write. The possible column models were described in the previous chapters. Finally, an accurate model of the competitive isotherms of the feed components is necessary. 

Let the volumetric hquid phase flow rates through the four columns be Qj, Qu, Qin, and Qjv, the flow rates at each node be Qd for the desorbent flow rate, Qe for the extract flow rate, Qf for the feed flow rate, and Qs for the raffinate flow rate. These flow rates are given by the following integral mass balance equations, written for both components at each node [27]. 

Compound Integration Mass Balance Donor time 0 Permeability ... 

Now we assume that the approximate solution, v , satisfies the integral mass balance relation Eq. 12.27, and we substitute Eq. 12.26 into Eq. 12.27 to obtain the following differential equation for the penetration front... 

If we substitute Eq. 12.36 into the integral mass balance equation (Eq. 12.6), we obtain... 

The integrated mass balance equation (13) expresses the fact that the local change of the mass of molecules across the film is compensated by the bulk and surface convection and diffusion fluxes. In the case of lubrication approximation for small Peclet numbers the solution of the leading order of the diffusion equations (4) and (6) gives the Boltzmann type of the nonequilibrium concentration distribution in the bulk phase ... 

The integrated mass balance for each type of ion gives the integration constant of the concentration profiles by writing the following ... 

Integrating eq.(8.4-7) with respect to time from 0 to t, and making use of the initial conditions (eq. 8.4-6) we obtain the following integral mass balance equation ... 

In order to derive the basic equation for a laminar or turbulent boundary layer, a small control volume in the boundary layer on a flat plate is used as shown in Fig. 3.10-5. The depth in the z direction is b. Flow is only through the surfacesand dj and also from the top curved surface at 8. An overall integral momentum balance using Eq. (2.8-8) and overall integral mass balance using Eq. (2.6-6) are applied to the control volume inside the boundary layer at steady state and the final integral expression by von Karman is (B2, S3)... 

The kinematic condition at the free interface can be replaced now by the integral mass balance in the film cross-section, which has a general form... 

A differential mass balance for a batch reactor, for constant density of the reaction mixture, is eq. (3.12), with Th integral mass balance is... 

Integral mass balance over the liquid phase... 

We start the derivation of the model by composing the unsteady integral mass balances for the system ... 

The principal mathematical feature shared by all steady-state packed-column operations is the distribution of the concentration variables in space, principally in the direction of flow. The model must therefore be composed of mass balances taken over a difference element of each phase, which are then converted into ODEs and integrated to obtain concentration profiles and other useful information. Algebraic (integral) mass balances also appear and can often be combined with the ODE balances to obtain important results. 

INTEGRAL MASS BALANCE OVER THE LIQUID PHASE... 

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