Binary systems mass balance

To date, in-bed filtration was more or less a black box as any measurement within the bed was virtually impossible. The design of such filters was based on models that could be validated only by integral measurements. However, with the MRI method even the (slow) dynamics of the filtration process can be determined. The binary gated data obtained by standard MRI methods are sufficient for the quantitative description of the system. With spatially resolved measurements the applicability of basic mass balances based on improved models can be shown in detail. 

If he selects the still pressure (which for a binary system will determine the vapour-liquid-equilibrium relationship) and one outlet stream flow-rate, then the outlet compositions can be calculated by simultaneous solution of the mass balance and equilibrium relationships (equations). A graphical method for the simultaneous solution is given in Volume 2, Chapter 11. 

For a simple distillation column separating a ternary system, once the feed composition has been fixed, three-product component compositions can be specified, with at least one for each product. The remaining compositions will be determined by colinearity in the ternary diagram. For a binary distillation only two product compositions can be specified independently, one in each product. Once the mass balance has been specified, the column pressure, reflux (or reboil ratio) and feed condition must also be specified. 

The flows in and out can be both convective (due to bulk flow) and molecular (due to dithision). We can write one eomponent continuity equation for each component in the system. If there are NC components, there are NC component continuity equations for any one system. However, the am total mass balance and these NC component balances are not all independent, since the sum of all the moles times their respective molecular weights equals the total mass. Therefore a given system has only NC independent continuity equations. We usually use the total mass balance and NC — 1 component balances. For example, in a binary (two-component) system, there would be one total mass balance and one component balance. 

The mass conservation equation only relates concentration variation with flux, and hence cannot be used to solve for the concentration. To describe how the concentrations evolve with time in a nonuniform system, in addition to the mass balance equations, another equation describing how the flux is related to concentration is necessary. This equation is called the constitutive equation. In a binary system, if the phase (diffusion medium) is stable and isotropic, the diffusion equation is based on the constitutive equation of Pick s law ... 

The preceding invited paper by Sahade (1987) gave a balanced overall picture for the subject matter. I shall attempt to complement his talk in discussing two classes of atmospheric diagnostics of the evolutionary processes in interacting binary systems. These are (I) Abnormal abundances of elements in the atmosphere resulting from the nuclear processes in the stellar interior. (II) Mass flow as a consequence of the evolution of either of the components and the Algol type binaries. 

A linear model predictive control law is retained in both cases because of its attracting characteristics such as its multivariable aspects and the possibility of taking into account hard constraints on inputs and inputs variations as well as soft constraints on outputs (constraint violation is authorized during a short period of time). To practise model predictive control, first a linear model of the process must be obtained off-line before applying the optimization strategy to calculate on-line the manipulated inputs. The model of the SMB is described in [8] with its parameters. It is based on the partial differential equation for the mass balance and a mass transfer equation between the liquid and the solid phase, plus an equilibrium law. The PDE equation is discretized as an equivalent system of mixers in series. A typical SMB is divided in four zones, each zone includes two columns and each column is composed of twenty mixers. A nonlinear Langmuir isotherm describes the binary equilibrium for each component between the adsorbent and the liquid phase. 

Figure 3.17 represents a continuous flow system where streams V andL are in thermodynamic equilibrium. This system can be evaluated by using thermodynamic equilibrium information with the appropriate number of mass balances. This will be illustrated first graphically and then analytically for a binary system. To analyze the system graphically, an equation must be obtained y =f(x) from mass balances (operating line) and plotted on an x-y equilibrium diagram. The intersection of the mass balance and equilibrium line is the solution. 

An equation is needed for Na to substitute into the above mass balance. For a binary system (A and B), the molar average velocity of flow (vm) for both components is... 

Because it is not continuous, the mathematical analysis of batch distillation is based on the total quantities. For a binary system in which the distillate is the desired product, the overall mass balance at the end of a batch run is... 

For a binary system with a single equilibrium stage, i.e., a reboiled stillpot with no rectification column, the mass balance is given by the Rayleigh equation ... 

The condensed phase density p, specific heat C, thermal conductivity A c, and radiation absorption coefficient Ka are assumed to be constant. The species-A equation includes only advective transport and depletion of species-A (generation of species-B) by chemical reaction. The species-B balance equation is redundant in this binary system since the total mass equation, m = constant, has been included the mass fraction of B is 1-T. The energy equation includes advective transport, thermal diffusion, chemical reaction, and in-depth absorption of radiation. Species diffusion d Y/cbfl term) and mass/energy transport by turbulence or multi-phase advection (bubbling) which might potentially be important in a sufficiently thick liquid layer are neglected. The radiant flux term qr... 

The solution of the Aim model is based on ihe steedy-state species balances (mass or molar units) that were developed in Section 2.3 for a binary system. 

Operating Line and "Equilibrium" Curve. Both terms are of importance for the graphical solution of a separation problem, i.e., for the graphical determination of the number of stages of a cascade. This method has been developed for the design of distillation columns by MacCabe and Thiele and should be well known. For all cases, the operating line represents the mass and material balances. In distillation, the equilibrium curve represents the thermodynamical va-por/liquid equilibrium. For an ideal binary system, the equilibrium curve can be calculated from Raoult s law and the saturation-pressure curves of the pure components of the mixture. In all other cases, however, for example, for all membrane processes, the equilibrium curve does not represent a thermodynamical equilibrium at all but will represent the separation characteristics of the module or that of the stage. 

For the case of isotopic exchange in a closed system, the increase of a heavy isotope by one phase must occur at the expense of the heavy isotope content of another, but the total amount of heavy isotope in such a system is constant (Criss 1999). The mass balance requirement for a binary system is given as... 

If we consider the system as a binary one with a surface-active material and bulk liquid, it is physically instructive to write the individual material balance relation for the surface excess concentration F (mol m ). The procedure for this is exactly as was carried out for the bulk binary system treated in Section 3.3. No chemical reaction at the interface is assumed, the system is considered to be dilute, the multicomponent mass flux is assumed to follow Pick s law, and the diffusion coefficients are taken to be constant. The expression for the surface concentration then becomes... 

Having obtained the proper constitutive flux equation (8.2-69) for this binary system, we now turn to obtaining the mass balance equation. Doing the mass balance across a thin element in the gas space above the liquid surface, we obtain the following mass balance equation at steady state ... 

We will now use the binary equilibrium data to develop graphical and analytical procedures to solve the combined equilibriurn, mass balance and energy balance equations. Mass and energy balances are written for the balance envelope shown as a dashed line in Figure 2-1. For a binary system there are two independent mass balances. The standard procedure is to use the overall mass balance,... 

The mass balances for batch distillation are somewhat different from those for continuous distillation. In batch distillation we are more interested in the total amounts of bottoms and distillate collected than in the rates. For a binary batch distillation, mass balances around the entire system for the entire operation time... 

Except for the limiting case of the irreversible isotherm discussed above the prediction of the temperature and concentration profiles requires the simultaneous solution of the coupled differential heat and mass balance equations which describe the system. The earliest general numerical solutions for a nonisothermal adsorption column appear to have been given almost simultaneously by Carter and by Meyer and Weber. These studies all deal with binary adiabatic or near adiabatic systems with a small concentration of an adsorbable species in an inert carrier. Except for a difference in the form of the equilibrium relationship and the inclusion of intraparticle heat conduction and finite heat loss from the column wall in the work of Meyer and Weber, the mathematical models are similar. In both studies the predictive value of the mathematical model was confirmed by comparing experimental nonisothermal temperature and concentration breakthrough curves with the theoretical curves calculated from the model using the experimental equilibrium... 

In industry many of the distillation processes involve the separation of more than two components. The general principles of design of multicomponent distillation towers are the same in many respects as those described for binary systems. There is one mass balance for each component in the multicomponent mixture. Enthalpy or heat balances are made which are similar to those for the binary case. Equilibrium data are used to calculate boiling points and dew points. The concepts of minimum reflux and total reflux as limiting cases are also used. 

In order to describe the dynamics of this system, we must include component material balances, an equation describing how the specific heat of the solution varies with the composition of the vessel, and the geometric relations which describe the heat-transfer area as a function of vessel volume. Since this is a binary mixture of components A and B, we can only write two material balances we choose to write one component balance and the overall mass balance. We will assume that the total mass density p is a constant for this system. Thus, the component mass balance is... 

Figure 7.6 shows the variables involved in a differential distillation process. For a binary system, they are four in number the moles liquid in the still or boiler at any instant W and its mole fraction Xg, the rate of vapor withdrawal D (mol/s), and the instantaneous vapor composition The mass balances... 

In contrast to the processes we encountered previously, which were largely or entirely isothermal in nature, distillation has substantial heat effects associated with it. Consequently, we expect heat balances to be involved in modeling the process, as well as the usual mass balances and equilibrium relations. These balances are formulated entirely in molar xmits because the underlying equilibrium relations, such as Raoult s law and its extension, or the separation factor a, are all described in terms of mole fractions. Thus, the flow rates L and G, which appear in Figure 7.16, are both in rmits of mol/s, enthalpies H in units of J/mol, and the liquid and vapor compositions are expressed as mole fractions x and y of a binary system. 

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