Fluid mean residence times

Note that the steady-state (/ - x) result depends only on the group (1 - E)VklF, the solid catalyst inverse space velocity-rate coefficient group, but the transient effects also require knowledge of (F /eF)", or the fluid mean residence time. 

Effect of Drum Speed on Solids and Fluid Mean Residence Times. Due to the difference in the axial velocity of the two phases (slip velocity) the mean residence time of the solid particles in the drum is in general higher than that of the fluid. The ratio of the mean residence time (t/Xj) is a measure of the slip velocity between... 

Effect of Drum Speed on t/Tj. Figure 32 displays the ratio of the solids and the fluid mean residence time as a function of the drum speed for various feed concentrations. For Cp = 7.8%, the solids to the fluid mean residence time ratio... 

Solids mean resistance time, s Xf Fluid mean residence time, s... 

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. 

Hoogendoorn and Lips (H10) carried out residence-time distribution experiments for countercurrent trickle flow in a column of 1.33-ft diameter and 5- and 10-ft height packed with -in. porcelain Raschig rings. The fluid media were air and water, and ammonium chloride was used as tracer. The total liquid holdup was calculated from the mean residence time as found... 

Frequently, stirred tanks are used with a continuous flow of material in on one side of the tank and with a continuous outflow from the other. A particular application is the use of the tank as a continuous stirred-tank reactor (CSTR). Inevitably, there will be a vety wide range of residence times for elements of fluid in the tank. Even if the mixing is so rapid that the contents of the tank are always virtually uniform in composition, some elements of fluid will almost immediately flow to the outlet point and others will continue circulating in the tank for a very long period before leaving. The mean residence time of fluid in the tank is given by ... 

Equation (1.40) is a special case of a far more general result. The mean residence time is the average amount of time that material spends in a flow system. For a system at steady state, it is equal to the mass inventory of fluid in the system divided by the mass flow rate through the system ... 

The terms space time and space velocity are antiques of petroleum refining, but have some utility in this example. The space time is defined as F/2, , which is what t would be if the fluid remained at its inlet density. The space time in a tubular reactor with constant cross section is [L/m, ]. The space velocity is the inverse of the space time. The mean residence time, F, is VpjiQp) where p is the average density and pQ is a constant (because the mass flow is constant) that can be evaluated at any point in the reactor. The mean residence time ranges from the space time to two-thirds the space time in a gas-phase tubular reactor when the gas obeys the ideal gas law. 

Example 3.5 A 1-in i.d coiled tube, 57 m long, is being used as a tubular reactor. The operating temperature is 973 K. The inlet pressure is 1.068 atm the outlet pressure is 1 atm. The outlet velocity has been measured to be 9.96 m/s. The fluid is mainly steam, but it contains small amounts of an organic compound that decomposes according to first-order kinetics with a half-life of 2.1s at 973 K. Determine the mean residence time and the fractional conversion of the organic. 

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 ... 

Consider the scaleup of a small, tubular reactor in which diffusion of both mass and heat is important. As a practical matter, the same fluid, the same inlet temperature, and the same mean residence time will be used in the small and large reactors. Substitute fluids and cold-flow models are sometimes used to study the fluid mechanics of a reactor, but not the kinetics of the reaction. 

The piston flow case assumes that the particles spend the same time in the reactor, i, even though the fluid phase is well mixed. This case resembles the mass transfer situation of piston flow in contact with a CSTR as considered in Section 11.1.4. The particles leave the reactor with size Ro — kf i. None will survive if f > Ro/k". Note that i is the mean residence time of the solid particles, not that of the fluid phase. 

Although the concept of mean residence time is easily visualized in terms of the average time necessary to cover the distance between reactor inlet and outlet, it is not the most fundamental characteristic time parameter for purposes of reactor design. A more useful concept is that of the reactor space time. For continuous flow reactors the space time (t) is defined as the ratio of the reactor volume (VR) to a characteristic volumetric flow rate of fluid (Y). 

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. 

For plug flow reactors all fluid elements take the same length of time to travel from the reactor inlet to the reactor outlet. This time is the mean residence time 7. 

This equation is the basic relation for the mean residence time in a plug flow reactor with arbitrary reaction kinetics. Note that this expression differs from that for the space time (equation 8.2.9) by the inclusion of the term (1 + SAfA) and that this term appears inside the integral sign. The two quantities become identical only when 5a is zero (i.e., the fluid density is constant). The differences between the two characteristic times may be quite substantial, as we will see in Illustration 8.5. Of the two quantities, the reactor... 

Levenspiel and Smith Chem. Eng. Sci., 6 (227), 1957] have reported the data below for a residence time experiment involving a length of 2.85 cm diameter pyrex tubing. A volume of KMn04 solution that would fill 2.54 cm of the tube was rapidly injected into a water stream with a linear velocity of 35.7 cm/sec. A photoelectric cell 2.74 m downstream from the injection point is used to monitor the local KMn04 concentration. Use slope, variance, and maximum concentration approaches to determine the dispersion parameter. What is the mean residence time of the fluid ... 

Mean residence time (f) is the average residence time of all elements of fluid in... 

The mean residence time, of fluid inside the vessel for steady-state flow is... 

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). 

Consider an element of fluid (as tracer ) entering the vessel at t = 0. Visualizing what happens to the element of fluid is relatively simple, but describing it quantitatively as Eft) requires an unusual mathematical expression. The element of fluid moves through the vessel without mixing with fluid ahead of or behind it, and leaves the vessel all at once at a time equal to the mean residence time ft = Vlq for constant density). Thus, Eft) = 0 for 0 < t < f, but what is Eft) at t = F ... 

This is also the fraction of the stream leaving vessel 2 that is of age between t and t + dt, since only the amount nUo entered at t = 0. But this fraction is also E(t)dt, from the definition of E(t). Thus, with V/q = t, the mean residence time for fluid in the two tanks (of total volume V), from (E),... 

Consider the entry of a small amount of fluid as tracer into the PFR at time t = 0. No tracer leaves the PFR until t = VPF/q0 = fPF, the mean residence time in the PFR, and hence no tracer leaves the two-vessel system, at the exit from the CSTR, during the period 0 sk fpF. As a result,... 

Asif et al. (1991) studied distributor effects in liquid-fluidized beds of low-density particles by measuring RTDs of the system by pulse injection of methylene blue. If PF leads into and follows the fluidized bed with a total time delay of 10 s, use the following data to calculate the mean-residence time and variance of a fluid element, and find N for the US model. 

The PFR model is based on turbulent pipe flow in the limit where axial dispersion can be assumed to be negligible (see Fig. 1.1). The mean residence time rpfr in a PFR depends only on the mean axial fluid velocity (U-) and the length of the reactor Lpfr ... 

If the mean residence time in the fluidised bed is sufficiently long, it may be regarded as a single stage, from which streams of fluid and solid leave in equilibrium. 

Consider a gas-phase reaction 2A R + 2S with unknown kinetics. If a space velocity of 1/min is needed for 90% conversion of A in a plug flow reactor, find the corresponding space-time and mean residence time or holding time of fluid in the plug flow reactor. 

The concentration readings in Table Ell.l represent a continuous response to a pulse input into a closed vessel which is to be used as a chemical reactor. Calculate the mean residence time of fluid in the vessel t, and tabulate and plot the exit age distribution E. 

Water is drawn from a lake, flows through a pump and passes down a long pipe in turbulent flow. A slug of tracer (not an ideal pulse input) enters the intake line at the lake, and is recorded downstream at two locations in the pipe L meters apart. The mean residence time of fluid between recording points is 100 sec, and variance of the two recorded signals is... 

VIV, reactor holding time or mean residence time of fluid in a flow reactor (s), see Eq. 5.24... 

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