Residence time distribution density

To measure a residence-time distribution, a pulse of tagged feed is inserted into a continuous mill and the effluent is sampled on a schedule. If it is a dry miU, a soluble tracer such as salt or dye may be used and the samples analyzed conductimetricaUy or colorimetricaUy. If it is a wet mill, the tracer must be a solid of similar density to the ore. Materials hke copper concentrate, chrome brick, or barites have been used as tracers and analyzed by X-ray fluorescence. To plot results in log-normal coordinates, the concentration data must first be normalized from the form of Fig. 20-15 to the form of cumulative percent discharged, as in Fig. 20-16. For this, one must either know the total amount of pulse fed or determine it by a simple numerical integration... 

What, if anything can be said about the residence time distribution in a nonisothermal (i.e., 7) / Tout) CSTR with variable density (i.e.. Pin Pout Rnd Qjfi Qout) ... 

Several age-distribution functions may be used (Danckwerts, 1953), but they are all interrelated. Some are residence-time distributions and some are not. In the discussion to follow in this section and in Section 13.4, we assume steady-flow of a Newtonian, single-phase fluid of constant density through a vessel without chemical reaction. Ultimately, we are interested in the effect of a spread of residence times on the performance of a chemical reactor, but we concentrate on the characterization of flow here. 

Consider the steady flow of fluid at a volumetric rate q through a stirred tank as a closed vessel, containing a volume V of fluid, as illustrated in Figure 13.4. We assume the flow is ideal in the form of BMF at constant density, and that no chemical reaction occurs. We wish to derive an expression for E(t) describing the residence-time distribution (RTD) for this situation. 

From the assumption of perfect mixing, the corresponding residence time distribution probability density function is well known as... 

The movement of the particles in this stage is very complex and extremely random, so that to determine accurately the residence time distribution and the mean residence time is difficult, whether by theoretical analysis or experimental measurement. On the other hand, the residence time distribution in this stage is unimportant because this subspace is essentially inert for heat and mass transfer. Considering the presence of significant back-mixing, the flow of the particles in this stage is assumed also to be in perfect mixing, as a first-order approximation, and thus the residence time distribution probability density function is of the form below ... 

It is noted that the right-hand side of Eq. (10.20) is just the series expansion of an exponential function. Therefore the overall residence time distribution probability density function in the SCISR is obtained to be... 

Impulse (delta) response method The input signal is changed in the form of a delta function. This method is widely used in chemical engineering to investigate the residence time probability density distribution function. 

Mixedness based on the residence time probability density distribution. 

Clear-liquor advance is used for two purposes (1) to reduce the quantity of liquor that must be processed by the solid-liquid separation equipment (e.g., filter or centrifuge) that follows the crystallizer, and (2) to separate the residence time distributions of crystals and liquor. The reduction in liquor flow through the separation equipment can allow the use of smaller equipment for a fixed production rate or increased production through fixed equipment. Separating the residence time distributions of crystals and liquor means that crystals will have an average residence time longer than that of the liquor. This should, in principle, lead to the production of larger crystals, but because the crystallizer is otherwise well mixed, the crystal population density will have the same form as that for the MSMPR crystallizer (Eq. (54)). 

However, if the reactor is filled, for example, with a catalyst, the situation becomes more complicated. The Vr would be the empty volume of the reactor, which is then difficult to determine, for instance, using settled apparent densities. The residence time can also be experimentally determined, usually resulting in a residence time distribution however, the experimental effort for such experiments is often large. Therefore, it is useful to apply a modified residence time, as shown in Equation (27), which defines the ratio of the mass of the catalyst and the gas flow, two easily measurable values ... 

Pressure Velocity Strain Viscosity Density Acceleration Gradients Tensor Volume Flow Force Melting Deformation Mixing Residence Time Distribution... 

The age of an atom or molecule in a reservoir is the time since it entered the reservoir. Age is defined for all molecules, whether they are leaving the reservoir or not. As with residence times, the probability density function of ages [ (r)] can have different shapes. In a steady-state reservoir, however, y>(r) is always a non-increasing function. The shapes of V(t) corresponding to the three residence time distributions discussed above are induded in Fig. 

In conventional fixed-bed reactors, catalyst particles of various sizes are often randomly distributed, which may lead to inhomogeneous flow patterns. Near the reactor walls, the packing density is lower than the mean value, and faster flow of the fluid near the wall is unavoidable. As a result, reactants may bypass the catalyst particles, and the residence time distribution (RTD) will be broadened. Moreover, the nonuniform access of reactants to the catalytic surface diminishes the overall reactor performance and can lead to unexpected hot spots and even to reactor runaway in the case of exothermic reactions. 

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