Transfer term

The charge transfer term arises from the transfer of charge (i.e. electrons) from occupied molecular orbitals on one molecule to unoccupied orbitals on the other molecule. This contribution is calculated as the difference between the energy of the supermolecule XY when this charge transfer is specifically allowed to occur, and an analogous calculation in which it is not. 

To increase the number of theoretical plates without increasing the length of the column, it is necessary to decrease one or more of the terms in equation 12.27 or equation 12.28. The easiest way to accomplish this is by adjusting the velocity of the mobile phase. At a low mobile-phase velocity, column efficiency is limited by longitudinal diffusion, whereas at higher velocities efficiency is limited by the two mass transfer terms. As shown in Figure 12.15 (which is interpreted in terms of equation 12.28), the optimum mobile-phase velocity corresponds to a minimum in a plot of H as a function of u. 

It is apparent from this expression that the larger the sum of chain transfer term becomes, the smaller will be... 

In the macroscopic heat-transfer term of equation 9, the first group in brackets represents the usual Dittus-Boelter equation for heat-transfer coefficients. The second bracket is the ratio of frictional pressure drop per unit length for two-phase flow to that for Hquid phase alone. The Prandd-number function is an empirical correction term. The final bracket is the ratio of the binary macroscopic heat-transfer coefficient to the heat-transfer coefficient that would be calculated for a pure fluid with properties identical to those of the fluid mixture. This term is built on the postulate that mass transfer does not affect the boiling mechanism itself but does affect the driving force. 

Likewise, the microscopic heat-transfer term takes accepted empirical correlations for pure-component pool boiling and adds corrections for mass-transfer and convection effects on the driving forces present in pool boiling. In addition to dependence on the usual physical properties, the extent of superheat, the saturation pressure change related to the superheat, and a suppression factor relating mixture behavior to equivalent pure-component heat-transfer coefficients are correlating functions. 

The first term on the right side of Eq. (5-179) is so nearly dominant for most furnaces that consideration of the main features of chamber performance is clarified by ignoring the loss terms and Lr or by assuming that they and have a constant mean value. The relation of a modified chamber efficiency T g(1 o) lo modified firing density D/(l — and to the normahzed sink temperature T = T-[/Tp is shown in Fig. 5-23, which is based on Eq. (5-178), with the radiative and convective transfer terms (GSi)/ja(TG — T ) -i- hiAijTc Ti) replaced by a combined radiation/conduction term (GS,) ,a(T - T ). where (GS])/ = (GS])/ + /jiA]/4oTgi Tg is adequately approximated by the arithmetic mean of Tg and T. ... 

Heat-Transfer Applications Heat-transfer analogs of common mass-transfer terms are ... 

When u E, this interstitial mixing effect was considered complete, and the resistance to mass transfer in the mobile phase between the particles becomes very small and the equation again reduces to the Van Deemter equation. However, under these circumstances, the C term in the Van Deemter equation now only describes the resistance to mass transfer in the mobile phase contained in the pores of the particles and, thus, would constitute an additional resistance to mass transfer in the stationary (static mobile) phase. It will be shown later that there is experimental evidence to support this. It is possible, and likely, that this was the rationale that explains why Van Deemter et al. did not include a resistance to mass transfer term for the mobile phase in their original form of the equation. 

It is now seen that only the resistance to the mass transfer term for the stationary phase is position dependent. All the other terms can be used as developed by Van Deemter, providing the diffusivities are measured at the outlet pressure (atmospheric) and the velocity is that measured at the column exit. 

The resistance to the mass transfer term for the stationary phase will now be considered in isolation. The experimentally observed plate height (variance per unit length) resulting from a particular dispersion process [e.g., (hs), the resistance to... 

It is a common procedure to assume certain conditions for the chromatographic system and operating conditions and, as a result, simplify equations (20) and (21). However, in many cases the assumptions can easily be over-optimistic, to say the least. It is necessary, therefore, to carefully consider the conditions that may allow such simplifying procedures and take steps to ensure that such conditions are carefully met when such expressions are used in practice. Now, the relative magnitudes of the resistance to mass transfer terms will vary with the type of columns (packed or capillary), the type of chromatography (GC or LC) and the type of particle, i.e., porous or microporous (diatomaceous support or silica gel). 

It is seen that the Van Deemter equation predicts that the total resistance to mass transfer term must also be linearly related to the reciprocal of the solute diffusivity, either in the mobile phase or the stationary phase. Furthermore, it is seen that if the value of (C) is plotted against 1/Dni, the result will be a straight line and if there is a... 

Figure 7. Graph of Resistance to Mass Transfer Term against the Reciprocal of the Solute Diffusivity in the Mobile Phase...

In Figure 7, the resistance to mass transfer term (the (C) term from the Van Deemter curve fit) is plotted against the reciprocal of the diffusivity for both solutes. It is seen that the expected linear curves are realized and there is a small, but significant, intercept for both solutes. This shows that there is a small but, nevertheless, significant contribution from the resistance to mass transfer in the stationary phase for these two particular solvent/stationary phase/solute systems. Overall, however, all the results in Figures 5, 6 and 7 support the Van Deemter equation extremely well. 

The predicted linear relationship between the resistance to mass transfer term and the square of the particle diameter is clearly demonstrated in Figure 8. The linear... 

Now, at high linear velocities, the longitudinal diffusion term will become insignificant and, equally important, the resistance to mass transfer term that incorporates the inverse function of diffusivity will become large, thus improving the precision of measurement. 

Now, it is of interest to determine if either the resistance to mass transfer term for the mobile phase or, the resistance to mass transfer term in the stationary phase dominate in the equation for the variance per unit length of a GC packed column. Consequently, taking the ratio of the two resistance to mass transfer terms (G)... 

Thus as (y) will always be greater than unity, the resistance to mass transfer term in the mobile phase will be, at a minimum, about forty times greater than that in the stationary phase. Consequently, the contribution from the resistance to mass transfer in the stationary phase to the overall variance per unit length of the column, relative to that in the mobile phase, can be ignored. It is now possible to obtain a new expression for the optimum particle diameter (dp(opt)) by eliminating the resistance to mass transfer function for the liquid phase from equation (14). 

Thus, the resistance to mass transfer term for the static mobile phase will be... 

It is seen that the peak capacity realized is far less than would be expected from the approximate calculation. This, in fact, is not surprising due to the size of the solute molecules. The diffusivity of the large solute molecules in either phase is so low that the resistance to mass transfer terms become inordinately large. Consequently, when operating significantly above the optimum velocity, very poor efficiencies are obtained. 

The heat transfer term envisions convection to an external surface, and U is an overall heat transfer coefficient. The heat transfer area could be the reactor jacket, coils inside the reactor, cooled baffles, or an external heat exchanger. Other forms of heat transfer or heat generation can be added to this term e.g, mechanical power input from an agitator or radiative heat transfer. The reactor is adiabatic when 7 = 0. 

The existence of three steady states, two stable and one metastable, is common for exothermic reactions in stirred tanks. Also common is the existence of only one steady state. For the styrene polymerization example, three steady states exist for a limited range of the process variables. For example, if Ti is sufficiently low, no reaction occurs, and only the lower steady state is possible. If Tin is sufficiently high, only the upper, runaway condition can be realized. The external heat transfer term, UAextiTout — Text in Equation (5.28) can also be used to vary the location and number of steady states. 

Solution Equation (5.29) is unchanged. The heat transfer term is added to Equation (5.30) to give... 

Note that the accumulation and reaction terms are based on the volume of the liquid phase but that the mass transfer term is based on the working volume, V=Vi + Vg. The gas-phase balance is... 

The mass transfer term in this equation is indeterminate since ki 00 and a — Ug 0. The indeterminacy is overcome by using Equation (11.5). Thus,... 

The central difficulty in applying Equations (11.42) and (11.43) is the usual one of estimating parameters. Order-of-magnitude values for the liquid holdup and kiA are given for packed beds in Table 11.3. Empirical correlations are unusually difficult for trickle beds. Vaporization of the liquid phase is common. From a formal viewpoint, this effect can be accounted for through the mass transfer term in Equation (11.42) and (11.43). In practice, results are specific to a particular chemical system and operating mode. Most models are proprietary. 

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 V( the component mass balance equations become for volume Vl... 

The first of these factors pertains to the complications introduced in the rate equation. Since more than one phase is involved, the movement of material from phase to phase must be considered in the rate equation. Thus the rate expression, in general, will incorporate mass transfer terms in addition to the usual chemical kinetics terms. These mass transfer terms are different in type and number in different kinds of heterogeneous systems. This implies that no single rate expression has a general applicability. 

Experimental values for several systems are given by Cornell et al. (1960), Eckert (1963), and Vital et al. (1984). A selection of values for a range of systems is given in Table 11.3. The composite mass transfer term KGa is normally used when reporting experimental mass-transfer coefficients for packing, as the effective interfacial area for mass transfer will be less than the actual surface area a of the packing. 

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