Dimensionless Residence Time

F Cumulative residence time distribution Dimensionless Dimensionless... 

For evafuating the residence time 0g of gas in the froth, the volume of the froth is taken as Aohf, where the height of the froth hj is obtained by first determining effective froth density [Eq. (14-117)]. The dimensionless froth density is defined by... 

When Equation 9 is used in Equation 8 along with the relationships for the residence time distributions one obtains the following dimensionless particle size distributions for one- and two-tank systems. 

T = reactor residence time Z = dimensionless axial distance... 

The second CSTR has the same rate constant and residence time, but the dimensionless rate constant is now based on (n, )2 = 0.618a rather than on Uin- Inserting A t2(am)2 = = (0-5)(0.618) = 0.309 into Equation... 

FIGURE 4.3 Effect of recycle rate on the performance of a loop reactor. The dimensionless rate constant is based on the system residence time, t = V/Q. The parameter is q/Q. 

The dimensionless variance has been used extensively, perhaps excessively, to characterize mixing. For piston flow, a = 0 and for a CSTR, a = l. Most turbulent flow systems have dimensionless variances that lie between zero and 1, and cr can then be used to fit a variety of residence time models as will be discussed in Section 15.2. The dimensionless variance is generally unsatisfactory for characterizing laminar flows where > 1 is normal in liquid systems. 

The entire residence time distribution can be made dimensionless. A normalized distribution has the residence time replaced by the dimensionless residence time, X = t/t. The first moment of a normalized distribution is 1, and all the moments are dimensionless. Normalized distributions allow flow systems to be compared in a manner that is independent of their volume and throughput. For example, all CSTRs have the same normalized residence time distribution, W(x) = exp(—t). Similarly, all PFRs have f(r) = S(x — 1). 

Determine the dimensionless variance of the residence time distribution in Problem 15.1. Then use Equation (15.40) to fit the axial dispersion model to this system. Is axial dispersion a reasonable model for this situation ... 

This equation has two parameters t, the mean residence time (z = V/F) with dimensions of time and k, the reaction rate constant with dimensions of reciprocal time, applying for a first-order reaction. The concentration of reactant A in the reactor cannot, under normal circumstances, exceed the inlet feed value, Cao and thus a new dimensionless concentration, Cai, can be defined as... 

The residence time distributions can be measured by applying tracer pulses and step changes as explained in Sec. 3.2.9. The response curves are best normalised such that the dimensionless time is... 

The above equations also apply to a plug flow reactor, where 0 is the dimensionless residence time, which varies with distance. 

The variance approach may also be used to determine n. From Illustration 11.2 the variance of the response data based on dimensionless time is 30609/(374.4)2, or 0.218. From equation 11.1.76 it is evident that n is 4.59. Thus the results of the two approaches are consistent. However, a comparison of the F(t) curves for n = 4 and n = 5 with the experimental data indicates that these approaches do not provide very good representations of the data. For the reactor network in question it is difficult to model the residence time distribution function in terms of a single parameter. This is one of the potential difficulties inherent in using such simple models of reactor behavior. For more advanced methods of modeling residence time effects, consult the review article by Levenspiel and Bischoff (3) and textbooks written by these authors (2, 4). 

Figure 11P.1 can be used to determine the dimensionless dispersion parmeter ( l/uL) for a system of interest. Use the transfer function method to evaluate the mean residence time and QjJuL) for a system subjected to the arbitrary input shown in the figure. Note that the output response has been shifted 62.5 sec to the left. Response values for the input and output streams were as follows. 

The parameter C in Eq. (25) is a dimensionless parameter inversely proportional to the average residence time of single particles on the heat transfer surface. It is suggested that this parameter be treated as an empirical constant to be determined by comparison with actual data in fast fluidized beds. The lower two dash lines in Fig. 17 represent predictions by Martin s model, with C taken as 2.0 and 2.6. It is seen that an appropriate adjustment of this constant would achieve reasonable agreement between prediction and data. 

The cumulative residence-time distribution function F(t) is defined as the fraction of exit stream that is of age 0 to t (i.e., of age t) it is also the probability that a fluid element that entered at t = 0 has left at or by time t. Since it is defined as a fraction, it is dimensionless. Furthermore, since F(O) = 0, that is, no fluid (of age 0) leaves the vessel before time 0 and F( ) = 1, that is, all fluid leaving the vessel is of age 0 to or all fluid entering at time 0 has left by time then... 

The holdback FI is the fraction of fluid within a vessel of age greater than t, the mean residence time. As a fraction, it is dimensionless. It can be obtained from age-distribution functions (see problem 13-4). 

Note that the dimensionless parameter PAR is the ratio of the residence time, L/v, and the reaction time for an nth-order reaction, 1 /kCA0. ... 

Figure 12.8 Calculated trajectories of fluid particles in the combustor with flame holder (solid lines) and the curves of constant dimensionless residence time t/tr (dashed curves). The residence time tr is defined as the time taken for the fluid particle to reach the turning point at the limiting trajectory (marked by the arrow). Conditions are similar to Fig. 12.66... 

Vazquez and Calvelo (1983b) presented a model for the prediction of the minimum residence time in a fluidized bed freezer which can then be equated to the required freezing time. The model is defined in terms of a longitudinal dispersion coefficient D, which is a measure of the degree of solids mixing within the bed in the direction of flow (and has the dimensions of a diffusivity, and hence units of m s ), a dimensionless time T... 

In many situations, it is convenient to plot a residence time distribution as a function of dimensionless time, 9, that is time normalised with respect to the mean residence time of the system under study. Thus... 

Any RTD curve, whether expressed in terms of dimensionless or real time, must retain the unit area property that is, all material passes through the system with a residence time somewhere between 0 and o°. Thus... 

Note that E(t) has dimensions of (time) and that E(0) is dimensionless. Intuitively, it is obvious that if a system has a mean residence time of, say, 2.3 min, then when the system RTD time scale is compressed by a factor of 1/2.3 (0 — t/2.3), the ordinate must be expanded by the reciprocal of this factor [E(0) = 2.3 E(f)] in order to maintain the unit area property. This is illustrated in Fig. 4. 

For the cumulative RTDs, the introduction of dimensionless time does not involve the same scaling factor and the fraction of material passing through the system with age less than 0 is the same as that fraction with a residence time less than the corresponding value of t. Thus... 

Now that the system mean residence time has been found to be 8.128 min, 6, defined by eqn. (2), and E(6), defined by eqn. (4), can be evaluated, thus enabling the system RTD to be presented in dimensionless form. These data are included in Table 2 appropriate calculations will confirm the unit area and unit mean properties of E(0). 

Equation (44) represents a family of RTDs all with a mean residence time of V/Q. As mentioned in Sect. 3.1, it is frequently more convenient to present RTDs in terms of dimensionless time, 6, rather than real time t. 

These dimensionless residence time aistributions E(0) can be obtained by inverting the fully normalised form of the system transfer function G(s). This is given by eqn. (45), which emphasises more clearly than eqn. (43) that the model in question contains only one flexible parameter, N. 

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