I&. E R ~Chem. . process ms. m v . i@a3,22,545-547
545
Measuring Local Flow Rates in Process Units Us1 Mann* Department of Chemical Englnmrlng, Texas Tech University, Lubbock, Texas 79409
Mlchael RublnovHch Facuity of Industrial Englnwlng and Management, Technion-Israel Institote of Technology, Haifa, Israel
An experimental method is descrlbed for measuring local flow rates and flow rate gradients inside continuous and batch process units. For continuous unlts, the method is based on placing a tagged (trace) particle in the inlet and measuring the number of times this particle visits the monltored zone as well the durations of these visits. These measurements are repeated for several particles. The local flow rate is readily calculated from the average number of visits by a particle to the zone. The volume of the monitored zone is calculated from the average duration of a visit. For batch units, the method is based on placing a single tagged particle in the system and measuring the number of visits to the monitored zone over a long (but known) period of time and the durations of these visits. A routine for calculating the gradient of the local flow rate is discussed. Methods for shortening the experiment time by placing several tagged particles simultaneously are discussed.
In a recent study of continuous flow systems (Rubinovitch and Mann, 1983) a relationship was derived tying together the local flow rate through a region and the number of visits to the region by a fluid element. Specifically, for a continuous system operating at steady state and for any specified zone inside the system (flow rate through the zone) = (net flow rate through the system) X (mean number of visits to the zone by a fluid element) In symbols wid= W o m l v 1
(1)
In this article we consider experimental techniques for measuring local internal flow rates in continuous process units by using the above relation. We also show that for closed (batch) systems a similar relation holds true and can also be used to measure local flow rates. Consider first a general continuous unit, at steady state, as shown schematically in Figure 1. Suppose we wish to determine the net flow rate through a specified zone of the system. The technique is based on placing a tagged particle in the inlet and installing a probe which detects and monitors the tagged particle whenever it is in the zone. For each tagged particle we monitor its successive visits to the zone before it exits the system. This is repeated with, say, m particles, and the average number of visits per particle to the zone is used to estimate E[W l m E [ M r - C ni (2) m i-1 where ni is the measured number of visits to the zone by the ith tagged particle. Since in most practical cases w o is known wld can be readily calculated from (1). This technique is especially suitable for determining local flow rates in particulate systems such as fluidized beds. However, it also can be used to determine local liquid flow rates by using a small tagged particle with the same density as the fluid. The specific tagging (or tracing) method (e.g., using radioactive particles) and the detection technique depend on the specific system, its dimensions, and operating conditions, as well as the fluid properties. We shall not discuss these points here. In the following,
we assume that the tagged particle has similar properties to the fluid elements and its movements represent the fluid flow. We note that in many applications it may be difficult to install a probe which covers exactly a predetermined space. The space (volume) which the probe monitors depends on numerous unknown parameters such as amount of tracing agent, sensitivity of the probe, fluid properties (e.g., transparency or absorption of tracer signal), and local flow conditions. Therefore, a method for determining the space which the probe monitors is called for. To this end, let us define the volume monitored as the zone for which the recorded signals is above a certain threshold level, say up Then estimating E[N (by (2)), we count only signals which give a reading above u1as shown schematically in Figure 2. On the other hand, the length of time intervals during which the signal is above u1 gives the durations of the successive visits to the zone. If we Y3,..., as shown in Figure 2, then denote these by Yl, Y2, the estimated mean duration of a visit to the zone is i
n
(3) where n is the total number of observed visit lengths. Recalling that V (4)
p=-
Wlocal
we can calculate the volume of the zone, V, since wlwd is already known. The number of times the measurement is repeated (number of tagged particles used) depends on the desired accuracy in determining wld and V. For any gi3en number of test particles, the accuracy of estimating E [ w and 1.1 can be evaluated using standard statistical t-tables (see, for example, Dixon and Massey, 1969). Then using trial and error, one can determine the needed value of m for a specified accuracy. Note that the method can be used to measure simultaneously local flow rates in several (even overlapping)zones by using several probes. Also, this method can be employed to determine the variation, with position, of the local flow rate as follows. Consider, for example, the
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Ind. Eng. Chem. Process Des. Dev., Vol. 22, No. 3, 1983 INLET
-
PARTICLE A
- ---
PARTICLE B
OUTLET
I
MONITORING ZONES
TIME, ARBITRARY SCALE
I
Figure 3. Schematic description of overlapping signals.
Figure 1. Schematic description of continuous system.
n I
i
of the number of visits to a zone, by a particle, prior to its departure from the system is not defined (or may be considered intinite) since fluid elementa never leave a batch system. In any case, eq 1,on which the previous method was based, does not apply. However, during any operating time one can measure, exactly as before, the number of times a tagged particle visits the region of interest as well as the duration of each visit. So what one needs is a relationship like (1)which ties together local flow rates and number of visits to a zone in a time interval. This can be derived as follows. Consider a certain local zone in an arbitrary but given batch system at steady state. Let N ( t ) be the number of times a tagged particle visits that zone, in time t. Let 1.1, be the mean time duration between two successive entries the following of the particle to the zone. Then, as t is true (see for example, Cinlar, 1975)
-
Q)
TIME ARBITRARY SCALE
Figure 2. Schematic description of measured signals.
measurements shown schematically in Figure 2. By selecting a series of sensitivity levels for the detector, ul, u2, ug,...,we obtain measurements of the number of visits and their durations for a series of local zones with corresponding volumes VI, V2, ..., as shown schematically in Figure 1. Since the volume of each zone and its flow rate are determined, one can readily calculate the local gradient of the flow rate. The main drawback of the technique described above is the long time needed to complete the experiments. This is particularly true when the zone of interest is small. To overcome the difficulty, one may choose to place simultaneously several tagged particles in the inlet and monitor all visits to the zone of interest until the last tagged particle leaves the system. In this case E [ N Jis estimated by
E[N z total number of visits monitored number of tagged particles used
However, this procedure may introduce some errors in the measurements which could happen when a monitored signal of one particle covers the signal of another particle as shown schematically in Figure 3. In order to minimize such errors it is perhaps best to feed tagged particles sequentially, either one at a time or a small number each time. We now turn to measurement of local flow rates in batch systems. Suppose that we waat to determine experimentally the flow rate through a specified local mne in a given batch system operating at steady state. The method used in continuous systems does not work here. The concept
Note that after each entry to the zone the particle resides for some time inside the zone and then leaves it and resides for a while in the remainder of the system before reentering the zone again. Let pl be the expected value of the duration of a single residence in the zone and p2 be the expected value of the time between departure and reentry to the zone. Clearly 1.1, = 1.11 + P2 (6) Also, if V is the volume of the zone and Vois the volume of the whole system, then V 1.11 = (74
1.12
=
v, - v
(7b)
since we assume the batch system is at steady state. Substituting (6) and (7)into (5), we obtain
and
which is the relation we need. Physically, (8) implies that the flow rate through a local zone in a batch system is given by the average number of visits by a fluid element to the
Ind. Eng. Chem. Process Des. Dev. 1983, 22, 547-552
zone, per unit time,-measured over a long time period, times the volume of the system. With this relation one can devise an experimental method for measuring local flow rates in batch systems. This can be done by using one tagged particle and monitoring its visits to the zone for a long time. Alternatively, one can monitor several tagged particles simultaneously but run a risk of introducing some errors in the measurements as discussed before. If m particles are used simultaneously and N J t ) is the total number of visits to the zone in time t , then the local flow rate is calculated by
Acknowledgment
Financial support for this work was made available by the Center for Energy Research, Texas Tech University, and by the Fund for the Promotion of Research at the Technion-Israel Institute of Technology. Nomenclature
E[Nl = mean number of visits to a region by a fluid element
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m = number of tagged particles used N ( t ) = number of times a fluid element visits a region in time t (in batch system) ni = number of visits to a region by the ith particle t = time u1,u2,... = sensitivity levels of the probes V0 = volume of the entire system V = volume of the zone w o = net flow rate through the system wl& = local flow rate Yi = duration of the ith visit to the region Greek Symbols pl = mean duration of a visit to the region p2 = mean duration between exit and reentry to the region pc = mean duration between two consecutive entries to the region (cycle time) L i t e r a t u r e Cited qinlar, E. “Introduction to Stochastlc Processes”; Prenticaliall, Inc.: Engiewood Cllffs, NJ, 1975; p 290. Dlxon, W. J.; Massey, F. J.. Jr. “Introduction to Statistical Analysis”. 3rd ed.; McQraw-Hill: New York, 1969 p 116. Rublnovbh, M.: Mann. U.. A I C M J . 1983, in press.
Received for review April 21, 1982 Accepted January 18, 1983
COMMUNICATIONS Sensltlvlty of Dlstillatlon Process Design and Operation to VLE Data Ail chemical process designs are based on inexact data and correlations. The effect of these inaccuracies on design can vary from insignificant to extremely critical. Design programs typically do not reveal the sensitivity of the important design variables to the uncertainty in the data, so that the design engineer must make inconvenient and expensive parametric studies to uncover the magnitude of the parametric sensitivity. This work presents a quick method for estimating the sensitivity of distlliation cdumn design to uncertainties in thermodynamic data. The estimation procedwe, which is based on short-cut design methods, is shown to provide accurate sensitivii estimates when compared to rigorous calculations. I t is proposed that the short-cut sensitivity procedure presented here be incorporated into computer-aided design programs to provide good estimates of parametric sensitivity at trivial cost for program coding and computer operating time.
Introduction
With the development of efficient computational algorithms the design and simulation of distillation towers is straightforward. These calculations are, however, complex (expensive) for multicomponent systems which consider fully nonidealities in the vapor-liquid equilibrium (VLE) Furthermore, the literature (Stocking et al., 1960; Kieffer, 1969; Zudkevitch, 1975; Williams and Albright, 1976; Mah, 1977; Sander, 1979; Streich and Kistenmacher, 1980) teaches that design of process equipment often is extremely sensitive to errors in physical properties, but it does not provide a convenient way to determine this sensitivity quantitatively. Of course, the designer always can repeat the entire process calculation with modified physical parameters, but this direct approach is not efficient for complex design methods, nor is it easily incorporated into process simulators. This note proposes the use of a specific set of short-cut calculation methods to estimate the relative response of
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distillation column variables to uncertainties or errors in VLE data. These relative responses are then used to couple the inexpensive short-cut methods with the accurate and expensive base case design to obtain good sensitivity estimates. The proposed method has been tested by comparing the results to rigorous calculations in three examples which span the range of industrial distillation design: (1)easy separation: butane-pentane splitter; (2) difficult separation: propene-propane superfractionator; (3) nonideal separation: methanol, ethanol, MEK, water. The tests were developed from two viewpoints: operation and design. In service the number of stages and the capacity of the tower are fixed. The operator then can attempt to meet product specifications which result from design errors due to uncertainties in VLE data by altering process variables. Two examples were considered: (a) changing the reflux ratio and the distillate rate to keep the recovery of the two key components constant and (b) changing the reflux ratio and the heavy key recovery to 0 1983 American Chemical Society
