Dimensionless flow rate

Friction factor, dimensionless Flow rate of one phase, GPM Aqueous phase flow rate, GPM Cy clone friction loss, expressed as number of cy clone inlet velocity heads, based on Drag or resistance to motion of body in fluid, poundals... 

Note that Figure 13 can be used to compare the parameters of the controller when they are obtained from the Ziegler-Nichols or Cohen-Coom rules. On the other hand, at Figure 14 it can be observed that the outlet dimensionless flow rate and the reactor volume reaches the steady state whereas the dimensionless reactor temperature remains in self-oscillation. The knowledge of the self-oscillation regime in a CSTR is important, both from theoretical and experimental point of view, because there is experimental evidence that the self-oscillation behavior can be useful in an industrial environment. 

However, due to limitation in xq dimensionless flow rate of cooling water it is necessary to consider two cases, depending on wherever xs is constrained or limited by the maximum opening of the control valve. 

Dimensionless bubble volumes predicted by solving Eq. (12-11) numerically and adding the second stage increment are plotted in Fig. 12.1 as functions of the dimensionless flow rate Q, with ju as parameter. It is important to note that ju is constant for a given orifice in a given gas-liquid system. Hence, Fig. 12.1 or Eqs. (12-10) to (12-12) give a convenient means of predicting bubble formation in any liquid-gas system whose properties are known. Over most of... 

Figure E3.6c plots the dimensionless flow rate q/qd, where qd is the drag flow rate, namely, the flow rate with zero pressure gradient, versus the dimensionless pressure gradient G. The figure shows that, whereas for Newtonian fluids, as expected, there is a linear relationship, non-Newtonian fluids deviate from linearity. The more non-Newtonian the fluid is, the greater is the deviation. Of particular interest is the inflection point indicating, for example, that in screw extruders, even for the isothermal case, increasing die resistance brings about somewhat unexpected changes in flow rate.

Fig. E3.6c Dimensionless flow rate versus dimensionless pressure gradient, with the Power Law exponent n as a parameter, for parallel-plate flow, as given in Eq. E3.6-24.

Fig. 9.6 Computed curves of dimensionless flow rate versus dimensionless pressure gradient for isothermal flow of a power law model fluid in shallow screw channels with the power law exponent n as a parameter, for helix angles 6f as follows O, 30° A, 20° , 10° solid curves are for a helix angle 30°. Note that for n < 1, the reduced flow rate is less than 1, with the deviation diminishing with decreasing of the helix angle. [Reprinted with permission from R. M. Griffith, Fully Developed Flow in Screw Extruders, Ind. Eng. Chem. Fundam., 1, 180-187 (1962).]...

For a generally applicable illustration, the computational results are made dimensionless (see Chapters 6 and 7). The dimensionless flow rate reads... 

A, Max. dimensionless flow rate at no pressure build-up, geometry parameter... 

Fig. 2. Response of the TAP reactor to an inlet pulse of a gas that is irreversibly adsorbed (or reacted) with a dimensionless rate constant k. Tp is the dimensionless time, and F p is the dimensionless flow rate. The model takes into account the number of molecules in the pulse A p A, the effective Knudsen diffusion coefficient DeA, the number of surface sites, and the dimensions of the reactor (after 55). A, = 0, standard diffusion curve B, = 3 C, U = 10.

An analogous procedure can also be applied to SMB processes (Tab. 7.5). The following compares the axial concentration profile of several 8-column SMB processes with different operating and design parameters but identical number of stages and dimensionless flow rate in each SMB section (m,) (Tab. 7.6). For this purpose we use... 

Storti et al. (1993b) have presented a similar approach by introducing the dimensionless flow rate ratio m,- as the ratio between liquid and solid flow in every section. 

According to these constraints the dimensionless flow rate ratios can be chosen. To achieve the best performance with respect to productivity, however, the difference between mn and mm should be as high as possible, representing the highest possible feed flow rate. The flow rate of fresh desorbent and, therefore, the eluent consumption can be minimized by choosing nq as low and mw as high as possible. 

To visualize all possible operating points for sections II and III, the dimensionless flow rate ratio mm is plotted versus mn in the so-called triangle diagram. For linear isotherms all possible operating points that fulfill the constraints are within the triangle shown in Fig. 7.18. 

To complete the set of possible operating parameters for a four-section SMB unit, values for the dimensionless flow rate ratios mt and mw have to be determined as well. In sections I and IV things are a little less complicated, since only the adsorption of single components is involved. 

In section IV the less retained component B has to be adsorbed and carried towards the raffinate port in order to regenerate the liquid phase. For the EMD53986 system with its multi-Langmuir isotherm, the corresponding constraint on the dimensionless flow rate ratio mIV is... 

These correlations can be transferred to continuous SMB or TMB chromatography where disperse as well as shock fronts are also present. The dimensionless flow rate ratios m,- can then be described as function of either the initial slope or the secant of the isotherm, depending on the situation in every zone. 

Here Nc stands for the total number of columns in the SMB plant. This procedure leads to higher pressure drops than in the real plant, since section I is the part of the process where, in general, the highest flow rate occurs. As demonstrated later, is not necessary to fix the flow rate in section I to the maximum allowed pressure drop to achieve optimal process performance. However, for a given flow rate in section I and all dimensionless flow rate ratios mj, the switching time can be calculated according to Eq. 7.78. 

In addition to the plate number the dimensionless flow rate ratios should remain constant. 

The direction of propagation of a component within the TMBR depends on the dimensionless flow rates in each section of the process. Assuming a linear isotherm a component propagates with the fluid if the dimensionless flow rate is higher than the Henry coefficient. If the flow rate is smaller than the Henry coefficient a component is transported in the direction of the solid flow. Therefore, Lode et al. (2001) subdivided the separation region into the three regions shown in Fig. 8.10. 

To calculate the conversion rate X of the reactant, the amount withdrawn by the product streams has to be known. As illustrated above, whether component A can reach the extract and/or the raffinate outlet depends on the chosen operating conditions. The ideal TMBR model and the resulting flow rates are shown in Fig. 8.7. Taking into account the external streams and dimensionless flow rates, X can be calculated by Eq. 8.29 ... 

Both the dimensionless velocity v and the dimensionless flow rate V can be written into single expressions ... 

The dimensionless flow rate is the actual flow rate divided by the drag flow rate, thus ... 

This is the same value as the optimum dimensionless flow rate described by Eq. 7.258. It is interesting to note that this throughput is determined only by the power law index. The data above the dashed line have been determined with Eq. 7.265. In this case, Fr < s + 1 and no extremum occurs in the velocity profile. The data below the dashed line have been determined with Eq. 7.262. In this case, Fr > s + 1 and an extremum does occur in the velocity profile. 

Figure 7.71 shows how the dimensionless flow rate changes with factor b. 

Figure 7.71 Dimensionless flow rate versus b for several values of the throttle ratio (r)...

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