Number of mass transfer units

The concept of a mass-transfer unit was developed many years ago to represent more rigorously what happens in a differential contactor rather than a stagewise contactor. For a straight operating line and a straight equilibrium line with an intercept of zero, the equation for calculating the number of mass-transfer units based on the overall raffinate phase N r is identical to the Kremser equation except for the denominator when the extraction factor is not equal to 1.0 [Eq. (15-23)]. 

The number of mass-transfer units N r is identical to the number of theoretical stages when the extrac tion fac tor is 1.0 [Eq. (15-24)]. When = 1.0,... 

Even staged eqmpment may be modeled best by the number of mass-transfer units when the extrartion fartor is much higher than 1.5, especially if the stage efficiencies are low. 

Heat may be transferred between two insoluble liquids in countercurrent flowthrough an extractor, and the performance can be evaluated in the same general manner as in mass transfer (Fig. 15-20). For a differential contactor the number of overall heat-transfer units based on the hot phase can be derived from the same equations used for the number of mass-transfer units based on the feed (raffinate) phase [Eq. (15-36)]. 

S02 removal is strongly correlated with the concentration of dissolved basic sulfite species in the scrubbing liquor. The enhancement effect of sulfite on the number of mass transfer units in a scrubber is expressed by a simple exponential relationship. 

Residence time distribution measurements, together with a theoretical model, provide a method to calculate the rate of mass transfer between the liquid flowing through the column, the dynamic holdup, and the stagnant pockets of liquid in between the particles. We have chosen the cross flow model (10). It has to be noted that the model starts from the assumption that the flow pattern has a steady-state character, which is in conflict with reality. Nevertheless, average values of the number of mass transfer units can be calculated as well as the part of the liquid being in the stagnant situation. 

Gravitational conversion l(kg-m)/(N-s) 4.18 EOS (Ibm- Number of mass transfer units Dimensionless Dimensionless... 

If the Zt- s of the previous equations are solved, they will be expressed in terms of the product of the reciprocal of the overbarred factors by the respective integrals. Thus, the tower height may be expressed as the product of two factors. Consider the first as the height of a mass transfer unit, H, and the second as the number of mass transfer units, N,. Therefore,... 

The difference becomes pronounced when values of the extraction factor are high. When E= 1, the number of mass-transfer units and number of theoretical stages are the same ... 

The above equation can be compared with Eq. 6.56, which was obtained with the lumped kinetic model. The two expressions are identical only if n = Nm, i-e. the average number of adsorption-desorption events in the microscopic model and the number of mass transfer units in the macroscopic kinetic model are analogous terms. Felinger et al. showed that not only the first and the second moments but also the whole band profiles obtained as the solutions of the microscopic model and the macroscopic lumped kinetic model are completely identical [9]. 

Figure 14.18 Concentration profile of an adsorbed solute along the column at different values of the rate constant. E = 10 D = O.OOlcm /s. Solid Une ka = 0.1 s , number of mass transfer units 250. Dotted line ka = 0.01 s, number of mass transfer units 25. Variable sample size. Reproduced from A. Jaulmes, C. Vidal-Madjar, Anal. Chem., 63 (1991) 1165 (Fig. 4). (c)1991 American Chemical Society.

The comparison of Eqs. 14.65 and 14.89 confirms that n = Nrea, i-e. the average number of adsorption-desorption steps (in the microscopic model) is identical to the number of mass transfer units (in the macroscopic model). 

When the number of transfer units is large, the band profiles calculated in overloaded elution with the solid film driving force model are very similar or identical to those calculated with the equilibrium-dispersive model [22]. By contrast, when the number of mass transfer units is small i.e., at very low values of kf), very different profiles can be obtained. 

The series of figures 16.6 to 16.9 illustrates the influence on their band profiles of the number of mass transfer units (k kf/k) of the column for the two components of a 1 9 binary mixture [22]. In Figure 16.6, this number is large (fcy i = kf 2 = 50 s ). The profiles obtained for the two components are the same as those obtained by numerical solution of the equilibrium-dispersive model (Chapter 11). Figure 16.7 shows the profiles obtained imder the same conditions, except for the value of the mass transfer coefficient kf i — kf 2 = 0.05 s ), resulting in 1,000 times fewer mass transfer units for the column. The change in band profiles from... 

Unusual band shapes are obtained when the number of mass transfer units is markedly different for the two components [22], For example. Figure 16.8 shows the chromatogram calculated with fcy j = 50 and fcj 2 = O-Ob s and Figure 16.9 the... 

The profiles of individual zones in displacement chromatography have also been calculated using the solid film linear driving force model [23]. Again, when the number of mass transfer units of the column is high, the results are very similar to those obtained with the equilibrium-dispersive model (Chapter 12). As an example. Figure 16.10 shows the displacement chromatogram calculated with kpi = kfg = = 50 s . The bands in the isotachic train are clearly formed... 

In order to calculate the number of mass transfer units, the mass transfer coefficients must be estimated, either based on experimental data or from empirical correlations. The following correlation (Chan and Fair, 1984) was developed for binary systems, and specifically applies to sieve trays. In this correlation Nq and N are expressed in terms of residence time and interfacial area instead of the standard forms of Equations 14.26 ... 

In the following developments, the number of mass transfer units is taken to be an independent variable and the compositions dependent variables. The equations for mass transfer on the plate of a distillation column are of the same form as those for a packed column. Thus, on the basis of the above assumptions, the component-material balance at any distance Z from the surface of the liquid on plate j may be represented by the following differential equation... 

Begin with the defining equations for the number of mass transfer units and the assumptions enumerated in the text and obtain the result given by Eq. (13-34), namely,... 

Number of mass-transfer units Nf, based on external film N, based... 

Define the concepts number of mass-transfer units (NTU), and height of a mass-transfer unit (HTU). 

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