Residence time average

1 The First Mmnent of E t)— The Average Residence Time Average Residence Time Consider the first moment of E(t) about the origin 

Suppose that fluid enters and leaves the reactor on/y by convection. A vessel that meets this criterion is known as a closed vessel. For this case, it can be shown that  

In this equation, V and v are, as usual, the reactor volume and the volumetric flow rate through the reactor, respectively. 

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Process development of the use of hydrogen as a radical quenching agent for the primary pyrolysis was conducted (37). This process was carried out in a fluidized-bed reactor at pressures from 3.7 to 6.9 MPa (540—1000 psi), and a temperature of 566°C. The pyrolysis reactor was designed to minimize vapor residence time in order to prevent cracking of coal volatiles, thus maximizing yield of tars. Average residence times for gas and soHds were quoted as 25 seconds and 5—10 rninutes. A typical yield stmcture for hydropyrolysis of a subbiturninous coal at 6.9 MPa (1000 psi) total pressure was char 38.4, oil... 

Samant and Ng [28] compared various scale-up rules for agitated reactors. They suggested that a scale-up rule of power per unit volume and constant average residence time (where the power per unit volume and average residence time cannot be increased) is the most suited in many operations. However, this still may not improve or preserve the performance of the systems. Therefore, adequate consideration must be given to a tradeoff between performance and operating constraints. 

Tlie average residence time of tlie vapor in tlie four foot section of pipe is ... 

Average residence time related to minimum residence time is ... 

Average residence time based on liquid flow... 

To provide reasonably adequate time for disengagement of foam and froth from the liquid in the downcomer, the total downcomer volume is checked against a minimum allowable average residence time of 5 seconds. 

ASpGr = Difference in specific gravity of the particle and the surrounding fluid tyvg = Average residence time based on liquid flow rate and vessel volume, min tmjp = Minimum residence time to allow particles to settle based on Stokes Law, min u = Relative velocity between particle and main body of fluid, ft/sec... 

Glaser and Lichtenstein (G3) measured the liquid residence-time distribution for cocurrent downward flow of gas and liquid in columns of -in., 2-in., and 1-ft diameter packed with porous or nonporous -pg-in. or -in. cylindrical packings. The fluid media were an aqueous calcium chloride solution and air in one series of experiments and kerosene and hydrogen in another. Pulses of radioactive tracer (carbon-12, phosphorous-32, or rubi-dium-86) were injected outside the column, and the effluent concentration measured by Geiger counter. Axial dispersion was characterized by variability (defined as the standard deviation of residence time divided by the average residence time), and corrections for end effects were included in the analysis. The experiments indicate no effect of bed diameter upon variability. For a packed bed of porous particles, variability was found to consist of three components (1) Variability due to bulk flow through the bed... 

They defined an average contact time as the average residence time of a bubble per unit length of dispersion. Their experimental data were correlated by ... 

Gal-Or and Resnick (G8) measured average residence time in a system that was geometrically similar to those used by Cooper et al (C9) and Yoshida et al (Y4) with air-distilled water and air-sodium sulfite solutions of the same concentration as used by these investigators. The ratio of impeller to tank diameter was 0.4 in one series (as in the work of Cooper and Yoshida) and 0.3 in a second series. Gal-Or and Resnick reported their results as an average residence time in seconds per foot of gas-free liquid, Bh. The average residence time was calculated from the equation... 

The same exponent was obtained in both works, although in one case a vaned-disk impeller was used, and in the other, arrowhead-shaped blades. Although the nominal residence time can be determined from a knowledge of gas holdup and volumetric gas flow rate, the nominal residence time (Eq. 36) will equal the statistical average residence time (Eq. 34) for certain cases only (L2). 

Under constant operating conditions, the number of bubbles in the vessel is constant, and their total volume or gas holdup is equal to Q0, where 0 is the average residence-time of the bubbles and Q is the volumetric gas flow rate. The number of bubbles in the vessel at any instant is then given by... 

Experimental work was undertaken (G8) to provide the information necessary to permit a test of this theoretical model. The system used bore complete geometrical and chemical similarity to that used by Cooper et al. (C9) so that their mass-transfer rate measurements, along with the average residence-time and power-consumption results determined in the experimental work (see Section II,D), were used to compare the experimental values with the model. 

Here is a modified average residence time defined by... 

The proposed technique will be used here to illustrate the case of interfacial heat and multicomponent mass transfer in a perfectly mixed gas-liquid disperser. Since in this case the holding time is also the average residence time, the gas and liquid phases spend the same time on the average. If xc = zd = f, then for small values of t, the local residence times tc and td of adjacent elements of the continuous and dispersed phases are nearly of the same order of magnitude, and hence these two elements remain in the disperser for nearly equal times. One may conclude from this that the local relative velocity between them is negligibly small, at least for small average residence times. Gal-Or and Walatka (G9) have recently shown that this is justified especially in dispersions of high <6 values and relatively small bubbles in actual practice where surfactants are present. Under this domain, Eqs. (66), (68), (69) show that as the bubble size decreases, the quantity of surfactants necessary to make a bubble behave like a solid particle becomes smaller. Under these circumstances (pd + y) - oo and Eq. (69) reduces to... 

This result can be useful for design purposes when the diffusivities, partition coefficients, feed-stream conditions, dispersed-system volume, gas-phase holdup (or average residence time), and the size distribution are known. When the size distribution is not known, but the Sauter-mean radius of the population is known, (293) can be approximated by... 

Lumped parameter propagation function Denoting difference Average residence time... 

Average residence time in reaction zone of a by-pass CFSTR... 

Pneumatic conveying dryers normally take particles 1-3 mm dia but up to 10 mm when the moisture is mostly on the surface. Air velocities are 10-30 m/sec. Single pass residence times are 0.5-3.0 sec but with normal recycling the average residence time is brought up to 60 sec. Units in use range from... 

X 10 years old, this implies that the content of the reservoir today is about half of what it was when the Earth was formed. The probability density function of residence time of the uranium atoms originally present is an exponential decay function. The average residence time is 6.5 x 10 years. (The average value of... 

It can be shown that for a reservoir in steady state. To is equal to t, i.e. the turnover time is equal to the average residence time spent in the reservoir by individual particles (Eriksson, 1971 Bolin and Rodhe, 1973). This may seem to be a trivial result but it is actually of great significance. For example, if tq can be estimated from budget considerations by comparing fluxes and burdens in Equation (1) and if the average transport velocity (V) within the reservoir is known, the average distance (L = Vxr) over which the transport takes place in the reservoir can be estimated. 

In the case of a human population corresponding to Fig. 4-4a, x is only about half of Tq. This example applies to the average age of all Swedes, which is around 40 years, whereas the average residence time, i.e., the average length of life (average age at death) is almost 80 years. 

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