Material balance equations, mass transfer

Material Balances Whenever mass-transfer applications involve equipment of specific dimensions, flux equations alone are inadequate to assess results. A material balance or continuity equation must also be used. When the geometiy is simple, macroscopic balances suffice. The following equation is an overall mass balance for such a unit having bulk-flow ports and ports or interfaces through which diffusive flux can occur ... 

The sign of the transfer term will depend on the direction of mass transfer. Assuming solute transfer again to proceed in the direction from volume VL to volume VG, the component material balance equations become for volume VL... 

In deriving the material balance equations, the dispersed plug flow model will first be used to obtain the general form but, in the numerical calculations, the dispersion term will be omitted for simplicity. As used previously throughout, the basis for the material balances will be unit volume of the whole reactor space, i.e. gas plus liquid plus solids. Thus in the equations below, for the transfer of reactant A kLa is the volumetric mass transfer coefficient for gas-liquid transfer, and k,as is the volumetric mass transfer coefficient for liquid-solid transfer. 

For planar or spherical electrodes, where the mass transport is a diffusion function in one dimension, it is possible to solve the diffusion equation as a function of time. In Section 3 the principles of how the cyclic voltammetric peak current could be calculated for a simple electron transfer reaction were presented. It is also possible to solve the material balance equations for the spherical electrode at steady state for a few first-order mechanisms (Alden and Compton, 1997a). In order to tackle second-order kinetics, more complex mechanisms, solve time-dependent equations or model other geometries with... 

Starting with the material balance equation, develop the expression for the impedance response for mass transfer through a stagnant film. 

The major part of the next few chapters are devoted to methods of estimating the low flux mass transfer coefficients k and [A ] and of calculating the high flux coefficients k and [/c ]. In practical applications we will need these coefficients to calculate the diffusion fluxes 7, and the all important molar fluxes N. The are needed because it is these fluxes that appear in the material balance equations for particular processes (Chapters 12-14). Thus, even if we know (or have an estimate of) the diffusion fluxes 7 we cannot immediately calculate the molar fluxes because all n of these fluxes are independent, whereas only n — 1 of the J I are independent. We need one other piece of information if we are to calculate the N. Usually, the form of this additional relationship is dictated by the context of the particular mass transfer process. The problem of determining the knowing the 7 has been called the bootstrap problem. Here, we consider its solution by considering some particular cases of practical importance. 

Since the rates of mass transfer across the liquid and vapor films must be equal at steady state (by material balance). Equations 15.9 and 15.10 may be combined to give the following ... 

The mass transfer equations discussed above are now combined with a material balance on the transferred component to calculate the column or packing height required for a given separation. The column cross-sectional area A is assumed known at this point although in a complete column design A must be determined based on pressure drop considerations. The column, which is in countercurrent flow with only liquid feed and vapor product at the top, and vapor feed and liquid product at the bottom (absorber, stripper, column section), is deflned as follows ... 

In the first case, mass transfer and reaction occur at different locations and are necessarily sequential—what reacts at the surface must first have got there by mass transfer. Mathematically, the equations for mass transfer are the same whether a reaction occurs or not. The reaction merely determines the boundary condition at the catalyst surface. In contrast, in the second case, mass-transfer and reaction occur side by side and simultaneously in the same volume elements. Here, mass-transfer enters as a source-or-sink term in the basic material-balance equation. 

Turning once more to the equations, we will derive code that will solve these numerically and simultaneously by using this expression for the saturation concentration of the salt and the linear dependence of density upon concentration. The code that follows does just this. The tank parameters are specified along with the volumes of the solution and salt phases at time zero (VIo and VIIo), the salt parameters, the mass transfer and flow rates, the maximum time for the integration to be done, the function calls for the exit flow rate in terms of the inlet flow rate, density of the solution and the saturation concentration of the salt, the material balance equations, the implementation of the numerical solution of the equations and the assignment of the interpolation functions to function names, and finally the graphical output routines. 

If the equilibrium curve and the material balance equations are expressed in other than the mass ratio concentrations X and Y and mass of solvent and mass of inert Fs and Ry, the operating line will not be straight, even under conditions of complete immiscibility, since the ratio of total phases is not constant because of transfer of solute between phases. 

Again, we consider a volume element of height dh. Mass transfer takes place at the wetted surface of the packing, and the effective area is difficult to estimate. Consequently, it is more convenient to discuss mass-transfer coefficients for a unit volume. We denote these as /(Tl for the liquid phase and Kq for the gas phase in units of kg mol m" hr" . Combining these definitions with the material balance equations above, we have... 

Equations for predicting constant-rate drying. Drying of a material occurs by mass transfer of water vapor from the saturated surface of the material through an air film to the bulk gas phase or environment. The rate of moisture movement within the solid is sufficient to keep the surface saturated. The rate of removal of the water vapor (drying) is controlled by the rate of heat transfer to the evaporating surface, which furnishes the latent heat of evaporation for the liquid. At steady state, the rate of mass transfer balances the rate ofheat transfer. 

Next, similar data for individual components of the system are to be found. One approach that has been often used in electrochemical investigations assumes that concentrations of components are close to the equilibrium ones when the system is sufficiently labile. Then expressions for the respective stability constants together with material balance equations are sufficient to obtain the required data. Though this approach sets unacceptable constraints on the EAC composition, it can be used as a satisfactory approximation when the rate constants of chemical steps exceed some critical values. Otherwise, the mass transfer problem should be solved without any simplifications accounting for the kinetics of chemical steps. 

If the T and P of a multiphase system are constant, then the quantities capable of change are the iadividual mole numbers of the various chemical species / ia the various phases p. In the absence of chemical reactions, which is assumed here, the may change only by iaterphase mass transfer, and not (because the system is closed) by the transfer of matter across the boundaries of the system. Hence, for phase equUibrium ia a TT-phase system, equation 212 is subject to a set of material balance constraints ... 

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