Equivalent stage, number

The column internals are either trays or packing. In a CS with trays, the vapor and liquid only come into contact at discreet points (i.e., on the trays). In a packed CS, there is continuous contact between vapor and liquid streams. The principle of height equivalent theoretical plate (HETP) is therefore used, which allows to model both packed and trayed CSs using the same position variable, n. Thus, the position variable, w, is often referred to as the number of trays or stage number. However, since it may represent a continuous packing rather than discreet points, neR. 

Not surprisingly, the fitness of strains for specific fermentation niches also has been found to vary. In some cases, only one or a few strains dominate throughout fermentation (Versavaud et al., 1995). In contrast, other researchers have found that different strains dominate at different stages of the fermentation (Sabate et al., 1998), or that several strains of Saccharomyces appear to be simultaneously present in equivalently high numbers (Torija et al., 2001 Vezinhet et al., 1992). Presumably, the biodiversity... 

Besides the dispersion model, the tanks in series model is the other onedimensional model widely used to represent non-ideal flow. Here the fluid is thought to flow through a series of equal-size ideal stirred tanks, and the parameter in this model is the number of tanks in the cascade (iV q). The RTD curves and moments of this model are easy to obtain, since problems of proper boundary conditions and method of tracer injection and measurement do not intrude. need not be an integer for curve-fitting purposes. It is strictly empirical, and no theoretical justification, such as Taylor diffusion, or theoretical estimates of the model parameter, are generally possible. This model starts from the mass balance equation for a series of i stirred vessels with 1 < i < iV, and AT, the number of vessels in the series (or the number of equivalent stages,... 

Both spray columns and packed columns routinely perform much less well than expected. There are two main reasons for this. First, the two liquid phases may not flow evenly past each other but mix more randomly together. In that case, they will behave more like a stirred tank than a packed bed, and have very few numbers of transfer units (NTU) or equivalent stages. Indeed, the fact that spray columns can show less mass transfer than even one equilibrium stage signals first how poor the contact between the two liquids can be. 

Many variations of the Morgan Algorithm were introduced, because of problems finding the terminating condition of stage 1 (oscillating number of equivalent classes [80]) or special atoms with isospcctral points [81],... 

Although the stagewise model is not physically reaUstic for differential contactors, it is sometimes used. The number of equivalent theoretical stages N can be determined graphically usiag the stepwise constmction illustrated ia Figure 7. For the case where both the equihbrium and operating lines are linear, it can be shown that ... 

Use of HETP Data for Absorber Design Distillation design methods (see Sec. 13) normally involve determination of the number of theoretical equihbrium stages or plates N. Thus, when packed towers are employed in distillation appRcations, it is common practice to rate the efficiency of tower packings in terms of the height of packing equivalent to one theoretical plate (HETP). 

The other common objective for calculating the number of countercurrent theoretical stages (or mass-transfer units) is to evaluate the performance of hquid-liquid extraction test equipment in a pilot plant or to evaluate production equipment in an industrial plant. Most liq-uid-hquid extraction equipment in common use can oe designed to achieve the equivalent of 1 to 8 theoretical countercurrent stages, with some designed to achieve 10 to 12 stages. 

The height equivalent to a theoretical stage (HETS) in an extraction tower is simply the height of the tower Zt divided by the number of theoretical stages achieved [Eq. (15-29)]. 

A related measure of efficiency is the equivalent number of stages erkngof CSTR battery with the same variance as the measured RTD. Practically, in some cases 5 or 6 stages may be taken to approximate plug flow. The dispersion coefficient also is a measure of deviation... 

If die equivalent number of perfectly mixed stages N is not an integer, let i be an integer slightly smaller dian N. A fraction 0 < h < 1 is defined as... 

An Excel spreadsheet (Example8-7.xls) was used to determine the various RTD functions and the computer program PROGS 1 was used to simulate the model response curve with the experimental data. The results show the equivalent number of ideally mixed stages (nCSTRs) for the RTD is 13.2. The Gamma distribution function from Equation 8-143 is ... 

Hengstebeck [137] presents a simplified procedure for reducing a multicomponent system to an equivalent binary using the key components. From this the number of stages or theoretical plates and reflux can be determined using conventional binary procedures and involving the McCabe-Thiele method. 

If the two representations are equivalent then Eqs. (3.79) and (3.80) describe how A s and B s must be transformed in terms of a s and /Ts. (These identities are performed explicitly by Sanchez and Di Marzio, [49]. Frank and Tosi [105] further show that if a s and /Ts are chosen to satisfy detailed balance conditions, that is equilibrium behaviour, then the occupation numbers of the two representations are only equivalent if the nv s are in an equilibrium distribution within each stage. This is likely to be true if there is a high fold free energy barrier at the end of each stem deposition, and thus will probably be a good representation for most polymers. In particular, the rate constant for the deposition of the first stem, A0 must contain the high fold free energy term, i.e. ... 

Clearly, the maximum degree of simplification of the problem is achieved by using the greatest possible number of fundamentals since each yields a simultaneous equation of its own. In certain problems, force may be used as a fundamental in addition to mass, length, and time, provided that at no stage in the problem is force defined in terms of mass and acceleration. In heat transfer problems, temperature is usually an additional fundamental, and heat can also be used as a fundamental provided it is not defined in terms of mass and temperature and provided that the equivalence of mechanical and thermal energy is not utilised. Considerable experience is needed in the proper use of dimensional analysis, and its application in a number of areas of fluid flow and heat transfer is seen in the relevant chapters of this Volume. 

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