Mean residence time

The first moment gives the average time the effluent molecules spent ui the rssdccr 

In previous chapters treating ideal reactors, a parameter frequently used was the space-time or average residence time x, which was defined as being equal to V/v. It will be shown that no matter what RTD exists for a particular reactor, ideal or nonideal, this nominal holding time, x, is equal to the mean residence time.,r . 

As is the case with other variables described by distribution frmctions, the mean value of the variable is equal to the first moment of the RTD fimc-tion, E(,t), Thus the mean residence time is 

Distributions of Residence Times for Chemical Reactors Chap. 13 

We now wish to show how we can determine the total reactor volinne using the cmnulative distribution function. Consider the following situation We have a reactor completely filled with maize molecules. At time / = 0 we start to inject blue molecules to replace the maize molecules that ciurently fill the reactor. That is, the reactor volimie V is equal to the volume occupied by the maize molecules. Now, in a time dt, the volume of molecules that will leave the reactor is vdt). The fraction of these molecules that have been in the reactor a time t or greater is [1 - T (f)]. Because only the maize molecules have been in the reactor a time t or greater, the volume of maize molecules, dV, leaving the reactor in a time dt is 

The concept of average residence time is defined here in a manner similar to the expected value in mathematical statistics  

time i can be determined by a numerical integration from t versus E(t) data  

Insertion of the time, t, into the equations implies that the counting of time starts from the time of injection. If h/Vr is denoted as c(0). Equation 4.12 transforms to a new form  

The C function is, consequently, defined as the ratio c(f)/c(0) or as a response to the impulse. By combining Equations 4.5 and 4.12, one is able to show that 

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The StabUity or persistence of a poUutant in the atmosphere depends on the poUutant s atmospheric residence time. Mean residence times and principal atmospheric sinks for a variety of species are given in Table 2. Species like SO2, (NO and NO2), and coarse particles have lifetimes less than... 

In 1973, a multistage surface-fermentation process was patented in Japan for the production of acetic acid (42) eight surface fermenters were connected in series and arranged in such a way that the mash passed slowly through the series without disturbing the film of yAcetobacter on the surface of the medium. This equipment is reported to produce vinegar of 5% acidity and 0.22% alcohol with a mean residency time in the tanks of 22 h. 

If the crystallizer is now assumed to operate with a cleat feed (n = 0), at steady state (dn jdt = 0), and if the crystal growth rate G is invariant and a mean residence time T is defined as then the population balance can be written as... 

As an idealization of the classified-fines removal operation, assume that two streams are withdrawn from the crystallizer, one corresponding to the product stream and the other a fines removal stream. Such an arrangement is shown schematically in Figure 14. The flow rate of the clear solution in the product stream is designated and the flow rate of the clear solution in the fines removal stream is set as (R — 1) - Furthermore, assume that the device used to separate fines from larger crystals functions so that only crystals below an arbitrary size are in the fines removal stream and that all crystals below size have an equal probabiHty of being removed in the fines removal stream. Under these conditions, the crystal size distribution is characterized by two mean residence times, one for the fines and the other for crystals larger than These quantities are related by the equations... 

The holdup of material in a continuous mill determines the mean residence time, and thus the extent of grinding. Gupta et al. [International J. Mineral Processing, 8, 345-358 (Oct. 1981)] analyzed published e)merimental data on a 40 X 40-cm grate discharge laboratoiy mill, ana determined that holdup was represented by = (4.020 — 0.176 WT)F , (0.040 0.01237 WT)S, - (4.970 0.395 Wl), where... 

The mean residence time T (defined as H JF) is the most important parameter, since it determines the time over which particles are exposed to grinding. Measurements on several industrial mills (Weller, Automation in Mining Mineral and Metal Processing, 3d IFAC Symposium, 303-309, 1980) (measured on the water, not the ore) showed that the maximum mill filhng was about 40 percent, and the maximum flow velocity through the mill is 40 m/h. 

One quantitative measure of reactor efficiency at a conversion level x is the ratio of the mean residence time or the reactor volume in a plug flow reactor to that of the reactor in question,... 

The chief quantities based on tracer tests are summarized in Table 23-4. Effluent concentrations resulting from impulse and step inputs are designated Cg and C , respectively. The initial mean concentration resulting from an impulse of magnitude m into a vessel of volume is C = mfVr- The mean residence time is the ratio of the vessel volume to the volumetric flow rate, t = V fV or t = jo tCg dt/jo Cg dt. The reduced time is t = t/t. 

Continuous. stirred tank reactor (CSTR), with the effluent concentration the same as the uniform vessel concentration. With a mean residence time t = V /V, the material balance is... 

The mean residence time is evaluated in terms of the fractional conversion as defined by Ca = Caq(1 X )-... 

The fractional conversion is determined from Equation 5-167 in terms of the mean residence time f = (V /u) to yield... 

Rearranging Equation 5-207 with the mean residence time t = Vj/u gives... 

Ideal continuous stiiTcd tank reactor (CSTRj behavior is approached when the mean residence time is 5-10 times the length needed to achieve homogeneity, which IS accomplished with 500-2,000 revolutions of a properly designed stiiTcr. 

Table 8-3 gives the parameters for calculating the mean residence time, variance, E(6), F(6), and 1(6) values. The mean residence time f with 6t = 2.5 is... 

Mean residence time The average time spent by the moleeules in a vessel. It is the first moment of the effluent eoneentration from a vessel with impulse input and is defined by... 

The dilution rate D is equal to 1/t, where t = u/V, and is equal to the mean residence time of the tank. Equation 11-63 shows that controlling the dilution rate D (Le., the flowrate into the reactor) can control the specific cellular growth rate of a microbial species p. 

Wewillintegrateover 0 andthenover t tosolvetheproblem.Thisisdoneinwhatfollows intwostepsbelow for clarity and with specific values forthe mean residence time andits deviationaboutthemean. 

The CSD from the continuous MSMPR may thus be predicted by a combination of crystallization kinetics and crystallizer residence time (see Figure 3.5). This fact has been widely used in reverse as a means to determine crystallization kinetics - by analysis of the CSD from a well-mixed vessel of known mean residence time. Whether used for performance prediction or kinetics determination, these three quantities, (CSD, kinetics and residence time), are linked by the population balance. 

Operating principle. Particles of terminal velocity Vi > mq will tend to settle therefore design for Vi < mq of the smallest particle present in the feed stream. In other words, the settling time should be less than the mean residence time of the up-flowing fluid. 

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