Mean residence number

The mean residence number (MRN) of a kinetic space is the average number of times the drug molecules enter a kinetic space. 

Example 3.4 Find the mean residence time in an isothermal, gas-phase tubular reactor. Assume that the reactor has a circular cross section of constant radius. Assume ideal gas behavior and ignore any change in the number of moles upon reaction. 

We are now ready to calculate the mean residence time. According to Equation (1.41), f is the ratio of mass inventory to mass throughput. When the number of moles does not change, t is also the ratio of molar inventory to molar throughput. Denote the molar inventory (i.e., the total number of moles in the tube) as Nactimi- Then... 

If the pilot reactor is turbulent and closely approximates piston flow, the larger unit will as well. In isothermal piston flow, reactor performance is determined by the feed composition, feed temperature, and the mean residence time in the reactor. Even when piston flow is a poor approximation, these parameters are rarely, if ever, varied in the scaleup of a tubular reactor. The scaleup factor for throughput is S. To keep t constant, the inventory of mass in the system must also scale as S. When the fluid is incompressible, the volume scales with S. The general case allows the number of tubes, the tube radius, and the tube length to be changed upon scaleup ... 

Via a passive scalar method [6] where or, denotes the volume fraction of the i-th phase, while T, represents the diffusivity coefiBcient of the tracer in the i-th phase. The transient form of the scalar transport equation was utilized to track the pulse of tracer through the computational domain. The exit age distribution was evaluated from the normalized concentration curve obtained via measurements at the reactor outlet at 1 second intervals. This was subsequently used to determine the mean residence time, tm and Peclet number, Pe [7]. 

The space time is not necessarily equal to the average residence time of an element of fluid in the reactor. Variations in the number of moles on reaction as well as variations in temperature and pressure can cause the volumetric flow rate at arbitrary points in the reactor to differ appreciably from that corresponding to inlet conditions. Consequently, even though the reference conditions may be taken as those prevailing at the reactor inlet, the space time need not be equal to the mean residence time of the fluid. The two quantities are equal only if all of the following conditions are met. 

The aforementioned investigators (10-12) have derived equations relating the measured mean residence times and variances to the Peclet number or dispersion parameter for the test section. For the case where the conditions at both monitoring probes correspond to a doubly infinite pipe, it can be shown that... 

Since the concentration profile given by Equation (2) is nearly symmetrical, it can be assumed that t occurs at the midpoint of the curve, therefore, this value is equivalent to the mean residence time, . For columns consisting of a large number of theoretical stages, the quantity (N-1)/N approaches unity and Equation (3) becomes... 

Calculate (a) the mean residence time of tracer in the vessel and (b) the dispersion number. If the reaction vessel is 0.8 m in diameter and 12 m long, calculate also the volume flowrate through the vessel and the dispersion coefficient. 

Level B utilizes the principles of statistical moment analysis. The mean in vitro dissolution time is compared to either the mean residence time or the mean in vivo dissolution time. Like correlation Level A, Level B utilizes all of the in vitro and in vivo data, but unlike Level A it is not a point-to-point correlation because it does not reflect the actual in vivo plasma level curve. It should also be kept in mind that there are a number of different in vivo curves that will produce similar mean residence time values, so a unique correlation is not guaranteed. 

A number of publications (6-10) have demonstrated that the size separation mechanism In HDC can be described by the parallel capillary model for the bed Interstices. The relevant expression for the separation factor, Rj., (ratio of eluant tracer to particle mean residence times) Is given by. 

Abbreviations. SJW, St. John s wort i.v., intravenous N, number of subjects C ax, maximum plasma concentration tnusx, time at Cmaxi h/2, elimination half life MRT, mean residence time. 

Numerous meteorites have been collected in Western Australia and terrestrial ages have been determined for 50 of them. Ages range from very young to around 40 000 years. There is a rough exponential decline in the number of meteorites as a function of age. The distribution of ages gives a mean residence time of 10 000 years at this location. 

Thousands of meteorites have also been recovered from the deserts of North Africa. Terrestrial ages of these meteorites extend out to 50 000 years and again the number of meteorites shows a rough exponential decrease with age. The mean residence time inferred for these meteorites is 12 000 years. 

Calculate the mean residence time of the fluid in the vessel and the dispersion number. 

Estimate (i) the mean residence time in the vessel and (ii) the value of the dispersion number for the vessel. 

Figure 5. Residence time distribution, characterised by the Pe number, expressed as a function of the mean residence time for rotor design A and B (viscosity of 600—800 P).

The distinction between these concepts is illustrated by the population of a country the average age of the population might be 40 years while the mean residence time, or life expectancy, might be twice that value. Human populations are not, however, a simple case for illustrating the concept of turnover time as we described it above, because they are not homogeneous in that all members do not have an equal probability of leaving at any time. If this population were homogeneous with respect to mortality and at steady state, turnover time would be the population divided by the number of members who die each year (the stock divided by the flux out). 

Comparison between Experimental Results and Model Predictions. As will be shown later, the important parameter e which represents the mechanism of radical entry into the micelles and particles in the water phase does not affect the steady-state values of monomer conversion and the number of polymer particles when the first reactor is operated at comparatively shorter or longer mean residence times, while the transient kinetic behavior at the start of polymerization or the steady-state values of monomer conversion and particle number at intermediate value of mean residence time depend on the form of e. However, the form of e influences significantly the polydispersity index M /M of the polymers produced at steady state. It is, therefore, preferable to determine the form of e from the examination of the experimental values of Mw/Mn The effect of radical capture mechanism on the value of M /M can be predicted theoretically as shown in Table II, provided that the polymers produced by chain transfer reaction to monomer molecules can be neglected compared to those formed by mutual termination. Degraff and Poehlein(2) reported that experimental values of M /M were between 2 and 3, rather close to 2, as shown in Figure 2. Comparing their experimental values with the theoretical values in Table II, it seems that the radicals in the water phase are not captured in proportion to the surface area of a micelle and a particle but are captured rather in proportion to the first power of the diameters of a micelle and a particle or less than the first power. This indicates that the form of e would be Case A or Case B. In this discussion, therefore, Case A will be used as the form of e for simplicity. 

Figure 5 represents a typical example of the variation of the number of polymer particles with mean residence time 0. The solid line shows the theoretical value predicted by the Nomura and Harada model with e= 1.28x 10 . The dotted line is that predicted by the Gershberg model(or the Nomura and Harada model with Case C for ), where Eq. (23) was used instead of Eq.(16) for Ap. The value of Nt produced at longer mean residence time differs, therefore, by a factor of T(5/3) between the solid and dotted lines in Figure 5. From the comparison between the experimental and theoretical results shown in Figure 5, it is confirmed that the steady state particle number can be maximized by operating the first stage reactor at a certain low value of mean residence time max which is considerably lower than that in the succeeding reactors. This is so-called "pre-reactor principle". It is, therefore, desirable to operate the first reactor at such mean residence time as producing something like a maximum number of polymer particles in order to increase the rate of polymerization in the succeeding reactors. This will result in a decrease in the number of necessary reactors and hence, in the capital cost.

Figure 5. Effect of mean residence time of the first reactor on the number of polymer particles formed (SF = 12.5 g/L H20 lF = 1.25 g/L H20 = 0.5...

This means that as long as a CSTR is used as the first stage reactor and all the recipe ingrediants are fed into the first stage reactor, one cannot have more than 57% of the number of particles produced in a batch reactor with the same recipe as in continuous operation. The validity of these expression is clear from the comparison between the experimental and theoretical values shown in Figure 5. From Figure 5, it is found that the optimum mean residence time of the first stage reactor is about 10 minutes under these reaction conditions. Equation(30) predicts 10.0 minutes, while experimental value is 10.4 minutes where the number of polymer particles is about 60% of that produced in a batch reactor. 

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