Equations liquid phase volume

Instead of the partial differential equation model presented above, the model is developed here in dynamic difference equation form, which is suitable for solution by dynamic simulation packages, such as Madonna. Analogous to the previous development for tubular reactors and extraction columns, the development of the dynamic dispersion model starts by considering an element of tube length AZ, with a cross-sectional area of Ac, a superficial flow velocity of V and an axial dispersion coefficient, or diffusivity D. Convective and diffusive flows of component A enter and leave the liquid phase volume of any element, n, as indicated in Fig. 4.24 below. Here j represents the diffusive flux, L the liquid flow rate and and Cla the concentration of any species A in both the solid and liquid phases, respectively. 

A comparison of equations (18.31) and (18.42) for the rninimum drop radius in horizontal and vertical separators shows that if the equality U /Uj(. = D/,/D holds (here the bottom indexes v and h correspond to the parameters of horizontal and vertical separators), then, all other things being equal, the minimum radii of drops in both separators coincide. But this certainly does not mean that their CE s should be the same, because the ratio of liquid phase volume concentrations at the exit of the two separators is... 

The estimation of Rav for characteristic parameter values shows that Rav where Aq = d/Re /" is the internal scale of turbulence. In a turbulent flow, both heat and mass exchange of drops with the gas are intensified, as compared to a quiescent medium. The delivery of substance and heat to or from the drop surface occurs via the mechanisms of turbulent diffusion and heat conductivity. The estimation of characteristic times of both processes, with the use of expressions for transport factors in a turbulent flow, has shown that in our case of small liquid phase volume concentrations, the heat equilibrium is established faster then the concentration equilibrium. In this context, it is possible to neglect the difference of gas and liquid temperatures, and to consider the temperatures of the drops and the gas to be equal. Let us keep all previously made assumptions, and in addition to these, assume that initially all drops have the same radius (21.24). Then the mass-exchange process for the considered drop is described by the same equations as before, in which the molar fluxes of components at the drop surface will be given by the appropriate expressions for diffusion fluxes as applied to particles suspended in a turbulent flow (see Section 16.2). In dimensionless variables (the bottom index 0" denotes a paramenter value at the initial conditions). 

The first variant [8] is based on the assumption that the solubility of initiator in the polymer is rather small and that is why it is displaced from the solid phase into the liquid phase via polymerization, in which its concentration is increased as a result of the decrease in its volumetric part Under this situation the relationship V =Vjy or v =v / -P) should exist between the specific rate of initiation Vi, determined by assuming that the initiator is uniformly distributed over the volume of the whole system, and the specific rate of initiation V y, under the assumption that all of the initiator is located in the liquid phase volume. Analysis shows, that in the presence of such a relationship the final equation, that will be equal to starting equation (5.1), cannot be obtained from equations (5.6) and (5.7). 

Evaluate ( ). The mixture volume V is determined from the equation of state, Eq. (4-231), applied to the liquid phase at the given composition, T, and P. 

Table 13-1, based on the binary-system activity-coefficient-eqnation forms given in Table 13-3. Consistent Antoine vapor-pressure constants and liquid molar volumes are listed in Table 13-4. The Wilson equation is particularly useful for systems that are highly nonideal but do not undergo phase splitting, as exemplified by the ethanol-hexane system, whose activity coefficients are snown in Fig. 13-20. For systems such as this, in which activity coefficients in dilute regions may...

Equation (2) is valid only for very dilute suspensions of nondeformable, smooth, uniform spheres. It assumes a Newtonian liquid phase and neglects interaction between particles, a plausible condition when the volume of the solid phase is small compared with the liquid phase. 

P the total pressure, aHj the mole fraction of hydrogen in the gas phase, and vHj the stoichiometric coefficient of hydrogen. It is assumed that the hydrogen concentration at the catalyst surface is in equilibrium with the hydrogen concentration in the liquid and is related to this through a Freundlich isotherm with the exponent a. The quantity Hj is related to co by stoichiometry, and Eg and Ag are related to - co because the reaction is accompanied by reduction of the gas-phase volume. The corresponding relationships are introduced into Eqs. (7)-(9), and these equations are solved by analog computation. 

Since Eqs. (5) and (6) are not restricted to the vapor phase, they can, in principle, be used to calculate fugacities of components in the liquid phase as well. Such calculations can be performed provided we assume the validity of an equation of state for a density range starting at zero density and terminating at the liquid density of interest. That is, if we have a pressure-explicit equation of state which holds for mixtures in both vapor and liquid phases, then we can use Eq. (6) to solve completely the equations of equilibrium without explicitly resorting to the auxiliary-functions activity, standard-state fugacity, and partial molar volume. Such a procedure was discussed many years ago by van der Waals and, more recently, it has been reduced to practice by Benedict and co-workers (B4). 

In their correlation, Chao and Seader use the original Redlich-Kwong equation of state for vapor-phase fugacities. For the liquid phase, they use the symmetric convention of normalization for y and partial molar volumes which are independent of composition, depending only on temperature. For the variation of y with temperature and composition, Chao and Seader use the equation of Scatchard and Hildebrand for a multicomponent solution ... 

The difficulties encountered in the Chao-Seader correlation can, at least in part, be overcome by the somewhat different formulation recently developed by Chueh (C2, C3). In Chueh s equations, the partial molar volumes in the liquid phase are functions of composition and temperature, as indicated in Section IV further, the unsymmetric convention is used for the normalization of activity coefficients, thereby avoiding all arbitrary extrapolations to find the properties of hypothetical states finally, a flexible two-parameter model is used for describing the effect of composition and temperature on liquid-phase activity coefficients. The flexibility of the model necessarily requires some binary data over a range of composition and temperature to obtain the desired accuracy, especially in the critical region, more binary data are required for Chueh s method than for that of Chao and Seader (Cl). Fortunately, reliable data for high-pressure equilibria are now available for a variety of binary mixtures of nonpolar fluids, mostly hydrocarbons. Chueh s method, therefore, is primarily applicable to equilibrium problems encountered in the petroleum, natural-gas, and related industries. 

The total flow rate, (Lmn + Gmn), leaving the mixer will be related to the total phase volumes VLm and Vcm by a hydrostatic equation, which will depend on the net difference in the head of liquid, between the levels in the mixer and in the settler. The actual form of this relationship might need to be determined experimentally, but could, for example, follow a simple, square-root relationship of the form, in which flow rate is proportional to the square root of the difference in liquid head, or indeed to the total volume of liquid in the mixer, e.g.. 

Consider a differential element of column volume, AV, height AZ and cross-sectional area, Ac, such that AV=Ac AZ. Component mass balance equations can be written for each of the liquid phases, where... 

Where is the initial analyte concentration in the liquid phase, C( the concentration of analyte in the gas phase, K the gas-liquid partition coefficient for the analyte at the analysis temperature, V, the volume of liquid phase, and V, the volume of gas phase (318-321,324,325). From equation (8.3) it can be seen that the concentration of the analyte in the headspace above a liquid in equilibrium with a vapor phase will depend on the volume ratio of the geis and liquid phases and the compound-specific partition coefficient which, in turn, is matrix dependent. The sensitivity 1 of the headspace sampling method can be increased in some instances adjusting the pH, salting out or raising the... 

Since the molar volume of a gas is much larger than the molar volume of the liquid phase, the Clapeyron equation can be reduced to... 

Equations similar to 12.3-10 to -15 may be written in terms of internal energy, U, with Cv, the heat capacity at constant volume, replacing CP. For liquid-phase reactions, the difference between the two treatments is small. Since most single-phase reactions carried out in a BR involve liquids, we continue to write the energy balance in terms of H, but, if required, it can be written in terms of U. In the latter case, it is usually necessary to calculate AU from AH and Cv from CP, since AH and CP are the quantities listed in a database. Furthermore, regardless of which treatment is used, it may be necessary to take into account the dependence of AH (or AU) and CP (or C,) on T ... 

For a liquid-phase reaction, or gas-phase reaction at constant temperature and pressure with no change in the total number of moles, the density of the system may be considered to remain constant. In this circumstance, the system volume (V) also remains constant, and the equations for reaction time (12.3-2) and production rate (12.3-6) may then be expressed in terms of concentration, with cA = nAlV ... 

Since this is a liquid-phase reaction, we assume density is constant The quantity V in the material-balance equation (14.3-5 or 15) is the volume of file system (liquid) in the reactor. The reactor (vessel) volume is greater than this because of file 75%-capacity requirement. From file specified rate law,... 

Thus in these equations for the steady-state, the convective terms are balanced by the transfer terms. Here the KLa value is based on the total volume Vx and is assumed to have the same value for each component. The linear velocities, expressed in terms of the volumetric flow rates and the empty tube cross-section, are for the gas and liquid phases... 

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