Residence time distribution equivalency

Axial Dispersion. Rigorous models for residence time distributions require use of the convective diffusion equation. Equation (14.19). Such solutions, either analytical or numerical, are rather difficult. Example 15.4 solved the simplest possible version of the convective diffusion equation to determine the residence time distribution of a piston flow reactor. The derivation of W t) for parabolic flow was actually equivalent to solving... 

In the statistical theory of fluid mixing presented in Chapter 3, well macromixed corresponds to the condition that the scalar means () are independent of position, and well micromixed corresponds to the condition that the scalar variances are null. An equivalent definition can be developed from the residence time distribution discussed below. 

The next thing we note about chromatography is that it is equivalent to tracer injection into a PFTR. Whereas in Chapter 8 we used tracer injection to determine the residence time distribution in a reactor, here we have nearly plug flow (with the pulse spread somewhat by dispersion), but adsorption from the fluid phase onto the solid reduces the flow velocity and increases the residence time to be much longer than x. ... 

Figure 17.1. Residence time distributions of some commercial and pilot fixed bed reactors. The variance, the equivalent number of CSTR stages, and the Peclet number are given for each.

Since the dimensionless time for a first-order reaction is the product of the reaction time t and a first-order rate constant k, there is no reason why k(x)t should not be interpreted as k(x)t(x), that is, the reaction time may be distributed over the index space as well as the rate constant. Alternatively, with two indices k might be distributed over one and t over the other as k x)t(y). We can thus consider a continuum of reactions in a reactor with specified residence time distribution and this is entirely equivalent to the single reaction with the apparent kinetics of the continuum under the segregation hypothesis of residence time distribution theory, a topic that is in the elementary texts. Three indices would be required to distribute the reaction time with a doubly-distributed continuous mixture. 

Any process takes a certain amount of time and the length of the residence time often dictates the occasions when particular equipment or technology can be used. On the other hand, in almost all chemical unit processes the driving forces vary from time to time, and therefore time has the nature of non-equivalence, i.e., an equal time interval yields different, even greatly different, results for the early and later stages of a process. The result mentioned here means the processing amount accomplished, such as the increments of reaction conversion, absorption efficiency, moisture removal etc. Normally, these parameters vary as parabolic curves with time. Because of the nature of the non-equivalence of time, in addition to the mean residence time, the residence time distribution (RTD) affects the performance of equipment, and thus receives common attention. 

The computer program PROG81 estimates the equivalent number of ideal tanks N for the given experimental residence time distribution... 

Allowable Spread in Residence Time. Other ways of stating the requirement of equal residence time of all parts of the reactant is that the flow through the reactor should approach plug flow or that the residence time distribution (RTD) should be equivalent to that in a large number of mixers in series. An often used rule of thumb is that this requirement is met when the equivalent number of mixers (N ) exceeds a certain value, say 5. However, this criterion is at best a semi-quantitative one, since the minimum value of is dependent upon the accepted deviation from the ideal reactor, and on the degree of conversion and the reaction order. 

Another scale-up variable that can be easily controlled is the length to diameter ratio L/D). Scale-up with an equivalent L/D ratio is beneficial in avoiding non-linear scale-up issues, especially at production scale. Residence time distribution (RTD) is a useful term for understanding scale-up of hot-melt extrusion processing. The RTD is used to attempt to quantify the average amount of time a material spends in the processor. The RTD depends on screw speed, screw element design, and material characteristics. The preference during scale-up is to maintain an equivalent ... 

Through similar representations of the residence time distribution of pipe flow reactors and ideally stirred tanks (CSTR) in series, Pawlowski [427] obtained a mutual association of the equivalent Bodenstein number BOeq = vL/DsSm of the plug flow reactor and the equivalent number Neq of the CSTR in series ... 

In order to improve the flow characteristics through the tubular reactor, 10 lengths (each 50 cm) of static mixer elements were inserted at 100 cm internals along the tube. Residence-time distribution studies using a pulse of potasiun chloride as tracer shewed that this tubular reactor had the flow characteristics equivalent to a cascade of 35 stirred tanks in series. Thus for practical purposes, this tube can be considered to have flow characteristics equivalent to those of a plug flow reactor. 

During this residence time distribution testing, the Photo-CREC-Water I was operated in the single pass mode with no water recycling and samples taken every 10 seconds. The Peclet number assessed with this method was 30 and the number of ideal CSTR tanks was estimated as 15. Since the number of baskets in Photo-CREC-Water I is 16, this demonstrates that each basket can be viewed as a close equivalent to an ideal single mixing stage. 

In a completely equivalent way one may determine the residence time distribution i(r) of site i and the probability rji that site i is occupied by a reactant molecule. 

The drawback of randomly packed microreactors is the high pressure drop. In multitubular micro fixed beds, each channel must be packed identically or supplementary flow resistances must be introduced to avoid flow maldistribution between the channels, which leads to a broad residence time distribution in the reactor system. Initial developments led to structured catalytic micro-beds based on fibrous materials [8-10]. This concept is based on a structured catalytic bed arranged with parallel filaments giving identical flow characteristics to multichannel microreactors. The channels formed by filaments have an equivalent hydraulic diameter in the range of a few microns ensuring laminar flow and short diffusion times in the radial direction [10]. 

Particle residence time distribution in a fluidized bed is more close to that of a stirred tank reactor (CSTR) or ideal backmix reactor than that of a plug flow reactor (Yagi and Kunii, 1961). In a perfect plug flow reactor, all particles have the same residence time, which is equivalent to the mean residence time of particles and can be calculated by... 

One engineering design area, i.e., impacted by nonideal reactor behavior and the accompanying fluid residence time distribution(s) is scaleup. Suppose that a reactor study is conducted at the pilot scale level and that the conversion (or the equivalent) associated with volume flow rate Qs are judged to be acceptable. The classical scaleup problem is to then design a larger process with flow rate qs which results in the same conversion. The scaleup factor SF is. 

In the method known as pulse, an amount of tracer is injected into the feed entering the reactor over a period of time approaching zero. The discharge concentration (or equivalent) is then measured as a function of time. Typical concentration curves at the outlet of the reactor can take the form of any of the residence time distribution plots discussed earlier. The most usual response takes the form similar of that in Figure 14.13. Generally, the response approaches a normal or log-normal distribution curve. 

F(t) is a probability distribution which can be obtained directly from measurements of the system s response in the outflow to a step-up tracer input in the inflow. Consider that at time t = 0 we start introducing a red dye at the entrance of the vessel into a steady flow rate Q of white carrier fluid. The concentration of the red dye in the inlet flow is C. At the outlet we monitor the concentration of the red dye, C(t . If our system is closed, i.e. if every molecule of dye can have only one entry and exit from the system (which is equivalent to asserting that input and output occur by convection only), then QC(t)/QCQ is the residence time distribution of the dye. This is evident since all molecules of the dye appearing at the exit at time t must have entered into the system between time 0 and time t and hence have residence times less than t. Only if our red dye is a perfect tracer, i.e.. if it behaves identically to the white carrier fluid, then we have also obtained the residence time distribution for the carrier fluid and F(t) = C(t)/C. To prove that the tracer behaves ideally and that the F curve is obtained, the experiment should be repeated at different levels of C. The ratio C(t)/C at a given time should be invariant to C, i.e. the tracer response and tracer input must be linearly related. If this is not the case, then C(t)/CQ is only the step response for the tracer, which includes some nonlinear effects of tracer interactions in the system, and which does not represent the true residence time distribution for the system. 

Since the same will happen in Reactor 2, in the end the ratio of polypropylene to ethylene-propylene copolymer per particle exiting Reactor 2 will also vary widely, which may be undesirable in some applications. Some of the reactor configurations shown in Figure 8.35 can reduce this phenomenon, particularly the configuration adopted for the gas-phase horizontal reactor, because the residence time distribution of this reactor is the equivalent to about three to four CSTRs in series. (Remember that the residence time of an infinite series of ideal CSTRs is that of a plug-flow reactor.) A more recent solution for this problem, in fact a completely new alternative to tandem reactor technology, is the multizone reactor that will be described in more detail below (see Section 8.6.4). 

In order to interpret the experimental residence time curves, comparison may be made with an ideal stirred tank cascade. Interpolation methods are known to get the number of equivalent tanks from the normalized residence time curves (8). This is only possible if the experimental residence time distribution is an ideal one. In many cases this is not true. Furthermore the residence time curves we get from our experimental arrangement comprise not only the residence time behavior of the bubble column itself but also the behavior of the liquid and gas distributors. These additional effects must not be neglected. 

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