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Phase transitions in physical systems have been an active area of scientific investigation for many years. Solid/liquid/vapor transitions such as those that occur between ice, water, and steam are undoubtedly familiar examples. Another common example is sugar crystal formation in honey that has been left to stand for a long time. Phase transitions, however, occur in many other types of systems—for example, the transition from ferromagnetic to paramagnetic behavior for ferromagnets at the Curie temperature, or the transition from normal conductivity to superconductivity in certain metals and ceramic oxides at a critical temperature, or the conductor/insulator percolation threshold for electrical networks.
Phase transitions are classified as being either first-order or continuous (or, more infrequently, second-order) phase transitions. First-order transitions such as those between ice, water, and steam, involve the evolution or absorption of heat at the transition point. Continuous transitions, by contrast, are not accompanied by heat transfer. In addition, continuous phase transitions are accompanied by the growth of fluctuations on ever-longer length scales. Transitions are further organized into groupings called universality classes. For systems from the same universality class, renormalizing raw experimental data to the proper critical conditions has the remarkable effect of “collapsing” data onto what is essentially a master curve.
Modeling of phase transitions is important for designing distillation columns using random or structured packings. In normal operation, packed columns are operated countercurrently with the vapor as the continuous phase and the liquid as the dispersed phase. These columns can reach points of hydraulic inoperability generally referred to as “flood points.” Flooding is typically associated with large fluctuations in the pressure drop, an abrupt increase in the liquid holdup and the pressure drop, and excessive liquid entrainment. Investigators have noted that flooding seems to be associated with a transition from vapor-continuous to liquid-continuous operation. See Eckert, J. S. New Look at Distillation-4 Tower Packings . . . Comparative Performance, Chem. Eng. Prog. 59 pp. 76-82 (May 1963).
The pressure drop of a vapor flowing countercurrently upward relative to the liquid flow is illustrated in
The flooding phenomenon in packed columns is extremely complex. It is possible to operate a column away from flooding in either vapor-continuous or liquid-continuous mode. The crossover from normal vapor-continuous operation to something more akin to operation in a liquid-continuous mode is signaled by the onset of a change in slope of the pressure drop versus vapor velocity from the slope of the curves away from flooding. In
For maximum product output from a distillation column, it is desirable to operate in vapor-continuous mode (with the gas as the vapor-continuous phase and the liquid as the dispersed phase) at the highest gas and liquid flow rates achievable without flooding the column. To assist in column design, packing manufacturers provide air/water pressure drop data at different liquid flow rates, such as the example shown in
The invention generally is directed to a method of determining a flood point for a packed column at any liquid flow rate, and a method of optimizing a product output rate from a distillation column by determining a pressure drop within the distillation column at a fraction of flood point by using a plot or mathematical expression of pressure drop as a function of fraction of flood point at any liquid flow rate.
The method of determining a flood point for a packed column includes providing a data set of gas pressure drop values as a function of gas flow rate values at several liquid flow rates through a packed column, and a known flood point value for one liquid flow rate. The method then includes setting flood point values for higher liquid flow rates at values lower than the known flood point value, and setting flood point values for lower liquid flow rates at values higher than the known flood point value, followed by expressing gas flow rates for liquid flow rates as fractions of the flood point value for each respective liquid flow rate. At a constant gas pressure drop, the method then includes calculating an average fractional flood point value for the liquid flow rates and minimizing the standard deviation between the fractional flood point value at different liquid flow rates and the calculated average fractional flood point value by iteratively resetting fractional flood point values and recalculating the average fractional flood point value for the liquid flow rates, thus resulting in determining a flood point for the packed column at any liquid flow rate, and thereby producing a plot of pressure drop as a function of fraction of flood point at any liquid flow rate, or a mathematical expression thereof.
In some embodiments, the packed column is operated with the gas as vapor continuous phase and the liquid as dispersed phase. In certain embodiments, the known flood point value for one liquid flow rate equals a gas pressure drop of about one and a half inches of water per foot of column for the gas being air and the liquid being water. In certain other embodiments, the known flood point value for one liquid flow rate equals a gas pressure drop of about two inches of water per foot of column for the gas being air and the liquid being water. In still other embodiments, the known flood point value for one liquid flow rate equals a gas flow rate at which a slope of gas pressure drop as a function of gas flow rate is equal to or greater than about two. Alternatively, the known flood point value for one liquid flow rate can include a user observation of an accumulation of liquid at the top of the column. In some embodiments, the method steps are computer-implemented and output is usable for process modeling.
The method of optimizing a product output rate from a distillation column having a top and a bottom includes setting a desired product output rate from the distillation column, calculating a fraction of flood point of the distillation column at a reflux ratio, and determining a pressure drop within the distillation column at the fraction of flood point. The step of determining the pressure drop employs the method of producing a plot of pressure drop as a function of fraction of flood point at any liquid flow rate, or producing a mathematical expression thereof. The method of optimizing a product output rate from a distillation column then includes calculating a pressure at the bottom of the distillation column for a pressure at the top of the distillation column, calculating the pressure drop within the distillation column of a given length, calculating a temperature corresponding to the calculated pressure at the bottom of the distillation column, and adjusting the desired product output rate or the reflux ratio or the pressure at the top of the distillation column. In some embodiments, the distillation column can be a packed column. The packed column is operated with gas as vapor continuous phase and liquid as dispersed phase.
An advantage of the methods described above is in the development of correlations to predict the pressure drop and approach to flood for a packing material. Instead of three variables—liquid flow rate, vapor (gas) flow rate, and pressure drop—the problem is reduced to one involving only two variables—fractional approach to flood and pressure drop. Then, a mathematical expression can be obtained for a curve fit through the data points. Such an expression can be used, for example, in a computer-implemented column design.
The foregoing will be apparent from the following more particular description of example embodiments of the invention, as illustrated in the accompanying drawings in which like reference characters refer to the same parts throughout the different views. The drawings are not necessarily to scale, emphasis instead being placed upon illustrating embodiments of the present invention.
A description of example embodiments of the invention follows. Presented in
The invention method 100 of determining a flood point for a packed column includes providing a data set of gas pressure drop values as a function of gas flow rate values at several liquid flow rates through a packed column, and a known flood point value for one liquid flow rate. An example using pressure drop and air/water flow rate data provided by the manufacturer (Koch-Glitsch LP, Wichita, Kans.) of the IMTP® #50 packing material will be used to illustrate the method 100.
The air/water flow rate data provided by the packing manufacturer is shown in
The method 100 (step 71) then includes setting flood point values for higher liquid flow rates, such as FSF30, at values lower than the known flood point value, and setting flood point values for lower liquid flow rates, such as FSF4, and FSH10, at values higher than the known flood point value. The initial flood point settings, FSF4, FSH10, and FSF30, for the other curves can be arbitrary (within the setpoint rule), such as, for example, the values at 2″ H2O/ft shown in
Then (at step 73), gas flow rates for liquid flow rates are expressed as fractions, χ, of the flood point value for each respective liquid flow rate as shown in Eq. 1 below.
(x)=(x4+x10+x20+x30)/4 (2)
At a constant gas pressure drop, the method then includes calculating an average fractional flood point value, <χ>, for the liquid flow rates, as shown in Eq. 2 above, and minimizing the standard deviation σ between the fractional flood point value at different liquid flow rates and the calculated average fractional flood point value. Steps 74-78 in
The overall procedure 100 as outlined above assumes that constant pressure drop data for several liquid loads are readily available. In terms of raw data this will rarely be true—experiments are usually not performed in a mode wherein pressure drops are held constant while vapor load is varied at constant liquid load. Instead, most experiments, as shown in
The invention method 100 iteratively resets fractional flood point values (step 701) and recalculates (loop 79) the average fractional flood point value for the liquid flow rates, thus resulting in determining a flood point for the packed column at any liquid flow rate, and thereby producing at 703 a plot of pressure drop as a function of fraction of flood point at any liquid flow rate, or a mathematical expression thereof.
The results for flood velocities of IMTP® #50 were:
FS, flood@ 4 gpm/ft2=3.6882 ft/sec[lb/ft3]1/2
FS, flood@ 10 gpm/ft2=3.13899 ft/sec[lb/ft3]1/2
FS, flood@ 20 gpm/ft2=2.66 ft/sec[lb/ft3]1/2
FS, flood@ 30 gpm/ft2=2.30067 ft/sec[lb/ft3]1/2
The method 100 described above defines a new criteria for flooding: the appropriate choice of flood points for a given set of packed column pressure drop data is the choice that results in the best possible collapse of the data onto a unified curve. The graph of the air/water pressure drop as a function of fraction of flood point for IMTP® #50 is shown in
Another example of using pressure drop and air/water flow rate data will be used to illustrate that the method 100 described above yields the best unified curve of pressure drop as a function of fraction of flood point, and produces flood point values that are the appropriate choice of flood points that are not necessarily the same values as the conventionally chosen flood point values. The air/water flow rate data for plastic Jaeger 2″ Tri-Packs® provided by the packing manufacturer is shown in
Table 1 shows that the flood point values obtained by the method of the invention are significantly different from the conventionally obtained values, particularly at lower liquid flow rates. In particular, the results shown in Table 1 indicate that at a low liquid flow rate (e.g., 6 gpm/ft2) the column will flood at a lower gas flow rate than expected, which could have negative consequences for column operation. By contrast, at high liquid flow rates (e.g., 40 gpm/ft2), the column output could be increased about 3% before flooding, a potentially significant increase in capacity.
y=−1.6555x5+3.6262x4−1.4727x3+1.2596x2−0.0533x+0.0161 0≦x≦1 (4)
where x is the fractional flood at constant liquid load and y is the dimensionless pressure drop. Such an expression can be used in a computer-implemented column design and process modeling enabled by the present invention.
The method of the present invention for optimizing a product output rate from a distillation column having a top and a bottom includes setting a desired product output rate from the distillation column. An example involving the separation of ethylbenzene from styrene monomer will be used to illustrate the method with reference to
The pressure P is related to the temperature T by Eq. 5:
where the A, B, and C coefficients are listed in Table 2 below.
Eq. 5 can be rearranged to obtain the temperature T as a function of pressure P as:
The vapor density can be obtained from Eq. (6) as:
For a pressure of 100 torr and a composition of almost entirely ethylbenzene at the top of the column 245, the EB vapor density is equal to 0.4895 kg/m3, at a temperature of 74.1° C. The corresponding liquid ethylbenzene density at that temperature is obtained from
The step 74 (
The four data points in Table 3 were fit to the Wallis equation
√{square root over (CSF)}+m29 {square root over (CLF)}=c (8)
where CSF is the gas flow rate and CLF is the liquid flow rate at the flood point. The following correlation parameters were obtained for air/water: cW=0.4152 (m/sec)1/2, mW=1.055. The fit and data points are shown in
Eq. 8 is then used to calculate a fraction of flood point of the distillation column at a reflux ratio assuming a constant molal overflow to complete a material balance around the column. The minimum reflux ratio can be obtained from the McCabe-Thiele diagram shown in
Solving Eq. 9 for Rmin yields a minimum reflux ratio of 6.88. Using the typical approximation that an acceptable reflux ratio is approximately 125% of Rmin yields a reflux ratio R=8.6. The McCabe-Thiele diagram for this separation is shown in
Turning back to
F=D+5.94 (10)
0.4F=0.98528D+(150×10−6)(5.94) (11)
where the mol % compositions have been converted to weight % for the purposes of this calculation. The results are F=9.998 kg/sec and D=4.058 kg/sec. The internal flows to and from the column 245 can now be estimated. By definition, the liquid rate returning to the top of the column 245 is
L
T
=R·D=(8.6)(4.058)=34.8988 kg/sec (11)
A mass balance around the top condenser shows that
V
T
=L
T
+D=34.8988+4.058=38.9568 kg/sec (12)
Since the feed 242 is assumed to be all liquid at its bubble point, the liquid rate exiting the column 245 must be the liquid rate at the top of the column plus the feed rate
L
B
=L
T
+F=9.998+38.9568=44.8968 kg/sec (13)
A mass balance around the bottom reboiler yields
V
B
=L
B
−B=44.8968−5.94=38.9568 kg/sec
The height of the column 245 is obtained by using Smoker's equation to calculate the number of theoretical stages required. Smoker's equation is strictly valid for systems with a constant relative volatility, α. The relative volatility for ethylbenzene relative to styrene does not vary greatly with composition, as shown in
m
T(α−1)kT2+[mT+bT(α−1)−α]kT+bT=0
where
Smoker's equation for the stripping (bottom) section of the column 245 is
m
B(α−1)kB2+[mB−bB(α−1)−α]kB+bB=0
Solving Eqs. 14 and 15 yields NT=31.687 and NB=67.021. The height equivalent to a theoretical plate (HETP) has been reported to be approximately 18″ (0.4572 m) for
Z=(NT+NB)(0.4572)=(98.708)(0.4572)≈45 meters (16)
In order to minimize cost and size, columns are normally run at about 80% of flood (at constant L/V), and therefore the column diameter will be estimated based on this criterion. In the great majority of cases columns first flood at the top. Therefore the diameter will be computed using conditions at the top of the column.
Solution of Eq. 17 yields
Dc=4.901 m≈4.9 m
The top of the column is operating at 80% of flood point (constant L/V). To determine the pressure drop at the top of the distillation column 245 at the fraction of flood point, the approach to flood point at constant liquid load is calculated from
From Eq. 18 above, the column 245 is operating at 0.77 of flood point. The corresponding pressure drop at the top of the distillation column 245 is obtained from
To estimate the vapor density at the bottom of the column 245, the pressure at the bottom of the column is initially estimated, for example, to be about 150 torr. The initial estimate can be modified and the calculation can be repeated if the pressure at the bottom of the column 245 is significantly different from the initial estimate. The vapor density at the bottom of the column 245 is obtained from Eq. 20 as
The density of liquid styrene as a function of temperature is show in
From
The pressure drop in the rectifying section is assumed to be everywhere the same as the pressure drop at the top of the column 245 and the pressure drop in the stripping section is assumed to be everywhere the same as that at the bottom of the column 245. Therefore, the pressure at the bottom of the column is:
This calculated pressure at the bottom of the column is approximately the same as the estimate used in Eq. 20, and is significantly less than the maximum bottom pressure of 192 torr. Therefore, this preliminary column design meets the column design requirements. Optionally, the desired product output rate, or the reflux ratio, or the pressure at the top of the column can be adjusted to operate the column at a pressure that is closer to the polymerization limit.
In one embodiment, the processor routines 92 and data 94 are a computer program product (generally referenced 92), including a computer readable medium (e.g., a removable storage medium such as one or more DVD-ROM's, CD-ROM's, diskettes, tapes, etc.) that provides at least a portion of the software instructions for the invention system. Computer program product 92 can be installed by any suitable software installation procedure, as is well known in the art. In another embodiment, at least a portion of the software instructions may also be downloaded over a cable, communication and/or wireless connection. In other embodiments, the invention programs are a computer program propagated signal product embodied on a propagated signal on a propagation medium (e.g., a radio wave, an infrared wave, a laser wave, a sound wave, or an electrical wave propagated over a global network such as the Internet, or other network(s)). Such carrier medium or signals provide at least a portion of the software instructions for the present invention routines/program 92.
In alternate embodiments, the propagated signal is an analog carrier wave or digital signal carried on the propagated medium. For example, the propagated signal may be a digitized signal propagated over a global network (e.g., the Internet), a telecommunications network, or other network. In one embodiment, the propagated signal is a signal that is transmitted over the propagation medium over a period of time, such as the instructions for a software application sent in packets over a network over a period of milliseconds, seconds, minutes, or longer. In another embodiment, the computer readable medium of computer program product 92 is a propagation medium that the computer system 50 may receive and read, such as by receiving the propagation medium and identifying a propagated signal embodied in the propagation medium, as described above for computer program propagated signal product.
Generally speaking, the term “carrier medium” or transient carrier encompasses the foregoing transient signals, propagated signals, propagated medium, storage medium and the like.
The relevant teachings of all patents, published applications and references cited herein are incorporated by reference in their entirety.
While this invention has been particularly shown and described with references to example embodiments thereof, it will be understood by those skilled in the art that various changes in form and details may be made therein without departing from the scope of the invention encompassed by the appended claims.
This application claims the benefit of U.S. Provisional Application No. 61/276,098, filed on Sep. 8, 2009. The entire teachings of the above application are incorporated herein by reference.
Number | Date | Country | |
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61276098 | Sep 2009 | US |