WO2025117117A1

WO2025117117A1 - Quantifying entrainment in a fluidized bed reactor

Info

Publication number: WO2025117117A1
Application number: PCT/US2024/053384
Authority: WO
Inventors: Gerardo Corona
Original assignee: ExxonMobil Technology and Engineering Co
Current assignee: ExxonMobil Technology and Engineering Co
Priority date: 2023-11-28
Filing date: 2024-10-29
Publication date: 2025-06-05
Anticipated expiration: 2026-05-28

Abstract

A method may include: obtaining two or more density measurements at two or more heights above a fluidized bed within a fluidized bed reactor, calculating an exponential decay constant using the two or more density measurements at the two or more heights; calculating an entrainment density using a reflux density model which has an input of the exponential decay constant; calculating an entrainment flux using an entrainment flux model which has an input of the entrainment density; and adjusting at least one operating parameter of the fluidized bed reactor in response to the calculated entrainment flux.

Description

QUANTIFYING ENTRAINMENT IN A FLUIDIZED BED REACTOR CROSS-REFERENCE TO RELATED APPLICATION [0001] This application claims the benefit of U.S. Provisional Application 63/603,501, filed November 28, 2023, entitled “Quantifying Entrainment in a Fluidized Bed Reactor”, the entirety of which is incorporated by reference herein. FIELD OF THE INVENTION [0002] This disclosure relates to fluidized bed reactors, and more particularly to methods of measuring and quantifying entrainment of polymer resin product in a gas phase polymerization reactor. BACKGROUND OF THE INVENTION [0003] In gas phase polymerization, a gaseous stream containing one or more monomers is passed through a fluidized bed under reactive conditions in the presence of a catalyst to produce a polymer resin product. As polymerization occurs a portion of the monomers are consumed, and the gas stream is heated in the reactor by the heat of polymerization. A portion of the gas stream exits the reactor and may be recycled back to the reactor with additional monomers and additives. As granules of polymer product are formed in the reactor, they must be removed or discharged in order to maintain a workable bed level as well as to obtain the desired commercial product. Measuring or quantifying polymer resin product entrainment in a fluidized bed reactor has been a challenging problem to solve. The use of static probes and/or acoustic probes in the cycle gas line of a fluidized bed reactor have been attempted but correlating the data to elevated entrainment events or cooler or plate fouling has not been successful. [0004] References of potential interest in this regard include: Brems et al., Modelling the transport disengagement height in fluidized beds, A^DV. P^OWDER T^ECH.155, 22:2 (March 2011); Kunii and Levenspiel, FLUIDIZATION ENGINEERING, Chapter 7 – Entrainment and Elutriation from Fluidized Beds, pg.17 (2d Ed.1991); US Patent Publication 2019/0184361; and US Pat No.7,985,811. SUMMARY OF THE INVENTION [0005] Disclosed herein is an example method including: obtaining two or more density measurements at two or more heights above a fluidized bed within a fluidized bed reactor; calculating an exponential decay constant using the two or more density measurements at the two or more heights; calculating an entrainment density using a reflux density model which has an input of the exponential decay constant; calculating an entrainment flux using an entrainment flux model which has an input of the entrainment density; and adjusting at least one operating parameter of the fluidized bed reactor in response to the calculated entrainment flux. [0006] Further disclosed herein is a method including: obtaining two or more density measurements at two or more heights above a fluidized bed within a fluidized bed reactor and inputting the two or more density measurements into a distributed control system (DCS); calculating, using the DCS, an exponential decay constant using the two or more density measurements at the two or more heights, an entrainment density using a reflux density model which has an input of the exponential decay constant, and calculating an entrainment flux using an entrainment flux model which has an input of the entrainment density; comparing, using the DCS, the calculated entrainment flux to a setpoint entrainment flux; and generating a control signal, using the DCS, and sending the control signal to the fluidized bed reactor to adjust at least one operating parameter of the fluidized bed reactor in response to the calculated entrainment flux such that the entrainment flux is closer to the setpoint entrainment flux. [0007] These and other features and attributes of the disclosed methods and systems of the present disclosure and their advantageous applications and/or uses will be apparent from the detailed description which follows. BRIEF DESCRIPTION OF THE DRAWINGS [0008] To assist those of ordinary skill in the relevant art in making and using the subject matter hereof, reference is made to the appended drawings, wherein: [0009] FIG.1 depicts a schematic of an illustrative gas phase polymerization system in accordance with certain embodiments of the present disclosure. [0010] FIG.2 is a graph of solids entrainment density for an illustrative gas phase polymerization system in accordance with certain embodiments of the present disclosure. [0011] FIG. 3 is a natural log versus height graph of solids reflux density for an illustrative gas phase polymerization system accordance with certain embodiments of the present disclosure. [0012] FIG.4a is an illustrative depiction of a gas phase reactor (Rx) with 3 differential pressure instruments in accordance with certain embodiments of the present disclosure. [0013] FIG.4b is an illustrative depiction of a gas phase reactor (Rx) with 3 nuclear densometer instruments in accordance with certain embodiments of the present disclosure. [0014] While the disclosed process and system are susceptible to various modifications and alternative forms, the drawing illustrates a specific embodiment herein described in detail by way of example. It should be understood, however, that the description herein of a specific embodiment is not intended to limit the invention to the particular forms disclosed, but on the contrary, the intention is to cover all modifications, equivalents, and alternatives falling within the spirit and scope of the invention as defined by the appended claims. DETAILED DESCRIPTION OF THE INVENTION [0015] Example embodiments are directed to gas phase polymerization reactors and, more particularly example embodiments disclose methods of measuring and quantifying entrainment of polymer resin product in a gas phase polymerization reactor. Gas-phase polymerization in a fluidized bed is an industrial process used in polymerizing monomers such as ethylene and ethylene comonomers to produce polyethylene polymer and copolymer compositions. It is generally known in the art that entrainment and carryover of a polymer product from the fluidized bed is problematic for a reactor system. The methods of quantifying the entrainment within the reactor, especially in the freeboard, are not robust, and oftentimes the calculated entrainment disagrees with observed entrainment in the form of cooler and plate fouling. The methods of measuring and quantifying entrainment of polymer resin product disclosed herein allows for an accurate real-time measurement and calculation of entrainment such that process variables can be changed and optimized in real time in response to measured entrainment. Further, the methods disclosed herein allow for control of entrainment through a chemical plant distributed control system (DCS). [0016] As discussed above, granules of polymer product are formed in the reactor, and they must be removed or discharged to maintain a workable bed level as well as to obtain the desired commercial product. This is preferably accomplished in a cyclic fashion wherein batches of granules are discharged at once. Since a typical gas phase reactor operates under pressurized conditions, such as 250, 290, 320, 350 psig or more, the process to discharge the granules must be performed by transferring the granules to a lower-pressure environment for processing into a commercial product. [0017] A portion of the polymer resin product may be entrained in the gas stream exiting the fluidized bed reactor. If elevated entrainment or bed carryover events are left unchecked, significant cooler and/or plate fouling can occur which may represent millions of dollars per year in lost production opportunity. Elevated entrainment/bed carryover and subsequent fouling may not be detected for a period of time after the event occurs. As such, fluidized bed reactors may be operated with a margin of safety such elevated entrainment and carryover is less likely to occur. [0018] Measuring or quantifying polymer resin product entrainment in a fluidized bed reactor has been a challenging problem to solve. The use of static probes and/or acoustic probes in the cycle gas line of a fluidized bed reactor have been attempted but correlating the data to elevated entrainment events or cooler or plate fouling has not been successful. Gas phase polymerization reactors may have an expanded section where particle velocity is reduced. The solids density at the widest point in the expanded section during operation is typically expected to be <1% of the upper fluidized bed bulk density (UFBD). which is on the order of < 0.5 lb/ft³ (8 kg/m³) The gas density at the widest point of the expanded section is on the order of 2.0 lb/ft³ (32 kg/m³). Reactor instrumentation lacks the data resolution needed to decouple gas density and particle entrainment changes, especially in an expanded section of the reactor. In a fluidized-bed reactor, the Fluidized Bulk Density (FBD) is usually measured both within the lower half and upper half of the bed. The measured density is referred to as Lower Fluidized Bulk Density or Upper Fluidized Bulk Density, depending on where the measurement is taken. Lower fluidized bulk density is typically measured from a bottom half of the fluidized bed such as between 0.5 feet (0.15 m) to 20 feet (6 m) above the distributor plate and upper fluidized bulk density is typically measured from a top half of the fluidized bed such as between 30 feet (2.4 m) to 50 feet (15.2 m) above the distributor plate, depending on the reactor size. The UFBD (upper fluid bed density) and/or LFBD (lower fluid bed density) are measured using any suitable density measurement technique such as nuclear densometers, differential pressure instruments (e.g., differential pressure taps using sensor leads), diaphragm sensors, or a combination thereof. [0019] There are several correlations in literature to calculate entrainment. However, these equations require a vast number of assumptions including particle density, wet vs dry particle, sphericity of particles, and particle clumping assumptions, among many other assumptions, each on which has a non-negligible contribution on the calculated entrainment. Oftentimes the calculated entrainment from a model does not agree with the observed entrainment for a given fluidized bed reactor. [0020] A more accurate determination of particle entrainment at different points within the fluidized bed reactor will be useful when creeping the process variables which are linked to production and entertainment. [0021] The words and phrases used herein should be understood and interpreted to have a meaning consistent with the understanding of those words and phrases by those skilled in the relevant art. No special definition of a term or phrase, i.e., a definition that is different from the ordinary and customary meaning as understood by those skilled in the art, is intended to be implied by consistent usage of the term or phrase herein. To the extent that a term or phrase is intended to have a special meaning, i.e., a meaning other than the broadest meaning understood by skilled artisans, such a special or clarifying definition will be expressly set forth in the specification in a definitional manner that provides the special or clarifying definition for the term or phrase. [0022] For example, the following discussion contains a non-exhaustive list of definitions of several specific terms used in this disclosure (other terms may be defined or clarified in a definitional manner elsewhere herein). These definitions are intended to clarify the meanings of the terms used herein. It is believed that the terms are used in a manner consistent with their ordinary meaning, but the definitions are nonetheless specified here for clarity. [0023] “C_n” as used herein, and unless otherwise specified, the term means hydrocarbon(s) having n carbon atom(s) per molecule, wherein n is a positive integer. [0024] As used herein, the indefinite article “a” or “an” shall mean “at least one” unless specified to the contrary or the context clearly indicates otherwise. Thus, embodiments using “an alpha- olefin” include embodiments where one, two or more alpha-olefins are used, unless specified to the contrary or the context clearly indicates that only one alpha-olefin is used. [0025] As used herein, “wt.%” means percentage by weight, “vol%” means percentage by volume, “mol%” means percentage by mole, “ppm” means parts per million, and “ppm wt” and “wppm” are used interchangeably and mean parts per million on a weight basis. All concentrations herein, unless otherwise stated, are expressed on the basis of the total amount of the composition in question. [0026] “Olefin,” alternatively referred to as “alkene,” is a linear, branched, or cyclic compound of carbon and hydrogen having at least one double bond. For purposes of this specification and the claims appended thereto, when a polymer or copolymer is referred to as “comprising” an olefin, the olefin present in such polymer or copolymer is the polymerized form of the olefin. For example, when a copolymer is described as having an “ethylene” content of 35 wt.% to 55 wt.%, it is understood that the mer unit in the copolymer is derived from ethylene in the polymerization reaction and the derived units are present at 35 wt.% to 55 wt.%, based upon the weight of the copolymer. [0027] “Polyethylene,” as used herein, means an ethylene homopolymer or a copolymer comprising at least 89 wt.% ethylene. The terms “polyethylene polymer,” “polyethylene,” “ethylene polymer,” “ethylene copolymer,” and “ethylene-based polymer” have the same meaning as polyethylene copolymer, except where otherwise indicated (e.g. where a polyethylene homopolymer is referred to, this means a polymer formed from ethylene monomer without comonomer units, e.g., 100 wt% ethylene-derived units). [0028] A “polyethylene grade” is a discrete polyethylene product having a consistent set of properties and is produced using the same catalyst and a unique set of polymerization conditions. “Polyethylene grade slate,” as used herein, means a discrete number of polyethylene products produced in a selected polymerization reaction zone, wherein each polyethylene product has a consistent set of properties and is produced using the same catalyst and a unique set of polymerization conditions. [0029] A “polymer” has two or more of the same or different repeating units/mer units or simply units. A “homopolymer” is a polymer having units that are the same. A “copolymer” is a polymer having two or more units that are different from each other. A “terpolymer” is a polymer having three units that are different from each other. The term “different” as used to refer to units indicates that the units differ from each other by at least one atom or are different isomerically. The definition of copolymer, as used herein, includes terpolymers and the like. Likewise, the definition of polymer, as used herein, includes homopolymers, copolymers, and the like. Furthermore, the terms “polyethylene copolymer”, “ethylene copolymer”, and “ethylene-based polymer” are used interchangeably to refer to a copolymer that includes at least 50 mol% of units derived from ethylene. [0030] Nomenclature of elements and groups thereof used herein are pursuant to the NEW NOTATION published in HAWLEYS CONDENSED CHEMICAL DICTIONARY, Thirteenth Edition, John Wiley & Sons, Inc., (1997) (reproduced there with permission from IUPAC), unless reference is made to the Previous IUPAC form noted with Roman numerals (also appearing in the same), or unless otherwise noted. [0031] “Operating temperature (Top),” as used herein, means the target operating temperature for the polymerization zone in a gas phase reactor to produce a desired grade of polyethylene. The operating temperature (T_op) is the target reactor temperature of within the set of polymerization conditions associated with the desired grade of polyethylene. The operating temperature (Top) is a threshold value below the kill temperature (Tk). Operating Temperature (Top) is the temperature at which the polymerization reaction is operated in order to prevent reaching the kill temperature (T_k). The threshold value can vary based on one or more of polyethylene grade, particular reactor configurations, and/or preference of the operator of a particular reactor. In some embodiments, the threshold value is 14°F (7.8°C), 15°F (8.3°C), 16°F(8.9°C), or 17°F (9.4°C). [0032] “Polymerization conditions,” as used herein, means conditions conducive to the reaction of one or more olefin monomers when contacted with an activated olefin polymerization catalyst to produce a polyolefin polymer, including a skilled artisan’s selection of temperature, pressure, reactant concentrations, optional solvent/diluents, reactant mixing/addition parameters, and other conditions within at least one polymerization reactor. [0033] “Reactor system,” as used herein, means the reactor and piping and equipment containing the circulating loop of cycle fluid, including, but not limited to, the cycle fluid heat exchanger. [0034] The term “nuclear radiation source” is used broadly to refer to any suitable generator of nuclear radiation. For example, “nuclear radiation source” can refer to radioisotopes that are sources of gamma-radiation, although the nuclear radiation source can also be the source of other types of radiation. In some embodiments, the nuclear radiation source may include caesium-137 (also written cesium-137) (¹³⁷Cs), cobalt-60 (⁶⁰Co), or combinations thereof. Some of the radioisotopes that are nuclear radiation sources are sources of gamma-radiation and other radiation, such as beta-radiation. [0035] The term “nuclear radiation detector” is used broadly to refer to any device capable of detecting nuclear radiation from one or more nuclear radiation sources. Nuclear radiation detectors may include, for example, devices capable of detecting alpha particles, beta particles, gamma rays, neutrons or any combinations thereof. The nuclear radiation detectors used in this disclosure are selected based on their suitability for detecting the radiation emitted from the nuclear radiation source. Polymerization Process [0036] FIG.1 depicts a flow diagram of an illustrative gas phase polymerization system 100 for making polymers, according to one or more embodiments. The polymerization system 100 can include a reactor 101 in fluid communication with one or more discharge tanks 155, compressors 170, and heat exchangers 175. The polymerization system 100 can also include more than one reactor 101 arranged in series, parallel, or configured independent from the other reactors, each reactor having its own associated discharge tanks 155, compressors 170, and heat exchangers 175, or alternatively, sharing any one or more of the associated discharge tanks 155, compressors 170, and heat exchangers 175. For simplicity and ease of description, the polymerization system 100 will be further described in the context of a single reactor train. [0037] Reactor 101 can include a cylindrical section 103, a transition section 105, and a velocity reduction zone or dome 107. The cylindrical section 103 is disposed adjacent to the transition section 105. The transition section 105 can expand from a first diameter that corresponds to the diameter of the cylindrical section 103 to a larger diameter adjacent the dome 107. As mentioned above, the location or junction at which the cylindrical section 103 connects to the transition section 105 is referred to as the “neck” or the “reactor neck” 104. Dome 107 has a bulbous shape. One or more cycle fluid lines 115 and vent lines 118 can be in fluid communication with the dome 107. Reactor 101 can include the fluidized bed 112 in fluid communication with the dome 107. The fluidized bed 112 can comprise solid particles, such as solid catalyst particles (supported or unsupported) and polymer particles (e.g., in various states of particle growth as polymerization reactions proceed). [0038] In general, the height to diameter ratio of the cylindrical section 103 can vary in the range of from about 2:1 to about 5:1. The range, of course, can vary to larger or smaller ratios and depends, at least in part, upon the desired production capacity and/or reactor dimensions. The cross-sectional area of dome 107 is typically within the range of from about 2 to about 3 multiplied by the cross-sectional area of the cylindrical section 103. [0039] The velocity reduction zone or dome 107 has a larger inner diameter than the fluidized bed 112. As the name suggests, the velocity reduction zone slows the velocity of the gas due to the increased cross-sectional area. This reduction in gas velocity allows particles entrained in the upward moving gas to fall back into the bed, allowing primarily only gas to exit overhead of reactor 101 through the cycle fluid line 115. The cycle fluid recovered via cycle fluid line 115 can contain less than about 10 wt%, less than about 8 wt%, less than about 5 wt%, less than about 4 wt%, less than about 3 wt%, less than about 2 wt%, less than about 1 wt%, less than about 0.5 wt%, or less than about 0.2 wt% of the particles entrained in fluidized bed 112. [0040] The reactor feed via line 110 can be introduced to the polymerization system 100 at any point. For example, the reactor feed via line 110 can be introduced to the cylindrical section 103, the transition section 105, the velocity reduction zone, to any point within the cycle fluid line 115, or any combination thereof. Preferably, reactor feed from line 110 is introduced to the cycle fluid in cycle fluid line 115 before or after the heat exchanger 175. In the Figure, the reactor feed via line 110 is depicted entering the cycle fluid in cycle fluid line 115 after the heat exchanger 175. The catalyst feed via line 113 can be introduced to the polymerization system 100 at any point. Preferably the catalyst feed via line 113 is introduced to the fluidized bed 112 within the cylindrical section 103. [0041] The cycle fluid via cycle fluid line 115 can be compressed in compressor 170 and then passed through heat exchanger 175 where heat can be exchanged between the cycle fluid and a heat transfer medium. For example, during normal operating conditions a cool or cold heat transfer medium via line 171 can be introduced to the heat exchanger 175 where heat can be transferred from the cycle fluid in cycle fluid line 115 to produce a heated heat transfer medium via line 177 and a cooled cycle fluid via cycle fluid line 115. In another example, during idling of the reactor 101 a warm or hot heat transfer medium via line 171 can be introduced to the heat exchanger 175 where heat can be transferred from the heat transfer medium to the cycle fluid in cycle fluid line 115 to produce a cooled heat transfer medium via line 177 and a heated cycle fluid via cycle fluid line 115. The terms “cool heat transfer medium” and “cold heat transfer medium” refer to a heat transfer medium having a temperature less than the fluidized bed 112 within reactor 101. The terms “warm heat transfer medium” and “hot heat transfer medium” refer to a heat transfer medium having a temperature greater than the fluidized bed 112 within reactor 101. The heat exchanger 175 can be used to cool the cycle fluid or heat the cycle fluid in the cycle fluid line 115, which in turn can cool or heat, respectively, the fluidized bed 112 as desired to achieve or maintain desired operating conditions of the polymerization system 100, e.g., during reactor start-up, normal operation, idling, and shut down. Illustrative heat transfer mediums can include, but are not limited to, water, air, glycols, or the like. It is also possible to locate the compressor 170 downstream from the heat exchanger 175 or at an intermediate point between several heat exchangers 175. [0042] After cooling, all, or a portion of the cycle fluid via cycle fluid line 115 can be returned to reactor 101. The cooled cycle fluid in cycle fluid line 115 can absorb the heat of reaction generated by the polymerization reaction. The heat transfer medium in line 171 can be used to transfer heat to the cycle fluid in cycle fluid line 115 thereby introducing heat to the polymerization system 100 rather than removing heat therefrom. The heat exchanger 175 can be of any type of heat exchanger. Illustrative heat exchangers can include, but are not limited to, shell and tube, plate and frame, U- tube, and the like. For example, the heat exchanger 175 can be a shell and tube heat exchanger where the cycle fluid via cycle fluid line 115 can be introduced to the tube side and the heat transfer medium can be introduced to the shell side of the heat exchanger 175. If desired, several heat exchangers can be employed, in series, parallel, or a combination of series and parallel, to lower or increase the temperature of the cycle fluid in stages. [0043] Preferably, the cycle gas via cycle fluid line 115 is returned to reactor 101 and to the fluidized bed 112 through fluid distributor plate (“plate” or “distributor plate”) 119, in a manner such that the solid particles of the fluidized bed 112 are maintained in a fluidized state. The plate 119 is preferably installed at the inlet to the reactor 101 to prevent polymer particles from settling out and agglomerating into a solid mass and to prevent liquid accumulation at the bottom of the reactor 101 as well to facilitate easy transitions between processes which contain liquid in the cycle fluid line 115 and those which do not and vice versa. Although not shown, the cycle gas via cycle fluid line 115 can be introduced into reactor 101 through a deflector disposed or located intermediate an end of the reactor 101 and the distributor plate 119. [0044] The catalyst feed via line 113 can be introduced to the fluidized bed 112 within reactor 101 through one or more injection nozzles in fluid communication with line 113. The catalyst feed is preferably introduced as pre-formed particles in one or more liquid carriers (i.e., a catalyst slurry). Suitable liquid carriers can include mineral oil and/or liquid or gaseous hydrocarbons including, but not limited to, propane, butane, isopentane, hexane, heptane octane, or mixtures thereof. A gas that is inert to the catalyst slurry such as, for example, nitrogen or argon can also be used to carry the catalyst slurry into reactor 101. In one example, the catalyst can be a dry powder. In another example, the catalyst can be dissolved in a liquid carrier and introduced into reactor 101 as a solution. The catalyst via line 113 can be introduced into reactor 101 at a rate sufficient to maintain polymerization of the monomer(s) therein. Hydrogen is added via line 114. [0045] Fluid via line 161 can be separated from a polymer product recovered via line 117 from reactor 101. The fluid can include unreacted monomer(s), hydrogen, induced condensing agents (ICAs), and/or inert materials. The separated fluid can be introduced to reactor 101. The separated fluid can be introduced to cycle fluid line 115. The separation of the fluid can be accomplished when fluid and product leave reactor 101 and enter the product discharge tanks 155 through valve 157, which can be, for example, a ball valve designed to have minimum restriction to flow when opened. Positioned above and below the product discharge tank 155 can be conventional valves 159, 167. Valve 167 allows passage of product therethrough. For example, to discharge the polymer product from reactor 101, valve 157 can be opened while valves 159, 167 are in a closed position. Product and fluid enter the product discharge tank 155. Valve 157 is closed, and the product is allowed to settle in the product discharge tank 155. Valve 159 is then opened permitting fluid to flow via line 161 from product discharge tank 155 to reactor 101. Valve 159 can then be closed and valve 167 can be opened and any product in the product discharge tank 155 can flow into and be recovered via line 168. Valve 167 can then be closed. Although not shown, the product via line 168 can be introduced to a plurality of purge bins or separation units, in series, parallel, or a combination of series and parallel, to further separate gases and/or liquids from the product. The particular timing sequence of the valves 157, 159, 167, can be accomplished by use of conventional programmable controllers which are well known in the art. [0046] Reactor 101 can be equipped with one or more vent lines 118 to allow venting the bed during start up, idling, and/or shut down. Reactor 101 can be free from the use of stirring and/or wall scraping. The cycle fluid line 115 and the elements therein (compressor 170, heat exchanger 175) can be smooth surfaced and devoid of unnecessary obstructions so as not to impede the flow of cycle fluid or entrained particles. [0047] The conditions for polymerizations vary depending upon the monomers, catalysts, catalyst systems, and equipment availability. The specific conditions are known or readily derivable by those skilled in the art. to about 140°C, often about 15°C to about 120°C, and more often about 70°C to about 110°C. Pressures can be within the range of from about 10 kPag to about 10,000 kPag, such as about 500 kPag to about 5,000 kPag, or about 1,000 kPag to about 2,200 kPag, for example. Catalyst Systems [0048] The term “catalyst system” includes at least one “catalyst component” and at least one “activator,” alternately at least one co-catalyst. The catalyst system can also include other components, such as supports, and is not limited to the catalyst component and/or activator alone or in combination. The catalyst system can include any number of catalyst components in any combination as described, as well as any activator in any combination as described. [0049] The term “catalyst component” includes any compound that, once appropriately activated, is capable of catalyzing the polymerization or oligomerization of olefins. Preferably, the catalyst component includes at least one Group 3 to Group 12 atom and optionally at least one leaving group bound thereto. The term “leaving group” refers to one or more chemical moieties bound to the metal center of the catalyst component that can be abstracted from the catalyst component by an activator, thereby producing the species active towards olefin polymerization or oligomerization. Suitable activators are described in detail below. [0050] As used herein, in reference to Periodic Table “Groups” of Elements, the “new” numbering scheme for the Periodic Table Groups are used as in the CRC Handbook of Chemistry and Physics (David R. Lide, ed., CRC Press 81^st ed.2000). [0051] As used herein, the terms “activator” refers to any compound or combination of compounds, supported or unsupported, which can activate a catalyst compound or component, such as by creating a cationic species of the catalyst component. For example, this can include the abstraction of at least one leaving group (the “X” group in the single site catalyst compounds described herein) from the metal center of the catalyst compound/component. Activators can include Lewis acids such as cyclic or oligomeric poly(hydrocarbylaluminum oxides) and so called non-coordinating activators (“NCA”) (alternately, “ionizing activators” or “stoichiometric activators”), or any other compound that can convert a neutral metallocene catalyst component to a metallocene cation that is active with respect to olefin polymerization. Illustrative Lewis acids include, but are not limited to, aluminoxane (e.g., methylaluminoxane “MAO”), modified aluminoxane (e.g., modified methylaluminoxane “MMAO” and/or tetraisobutyldialuminoxane “TIBAO”), and alkylaluminum compounds. Ionizing activators (neutral or ionic) such as tri (n- butyl)ammonium tetrakis(pentafluorophenyl)boron may be also be used. Further, a trisperfluorophenyl boron metalloid precursor may be used. Any of those activators/precursors can be used alone or in combination with the others. There are a variety of methods for preparing aluminoxane and modified aluminoxanes known in the art. [0052] The catalyst compositions can include a support material or carrier. As used herein, the terms “support” and “carrier” are used interchangeably and are any support material, including a porous support material, for example, talc, inorganic oxides, and inorganic chlorides. The catalyst component(s) and/or activator(s) can be deposited on, contacted with, vaporized with, bonded to, or incorporated within, adsorbed or absorbed in, or on, one or more supports or carriers. Other support materials can include resinous support materials such as polystyrene, functionalized or crosslinked organic supports, such as polystyrene divinyl benzene polyolefins or polymeric compounds, zeolites, clays, or any other organic or inorganic support material and the like, or mixtures thereof. [0053] Inorganic oxide supports can include Group 2, 3, 4, 5, 13 or 14 metal oxides. The preferred supports include silica, which may or may not be dehydrated, fumed silica, alumina, silica-alumina and mixtures thereof. Other useful supports include magnesia, titania, zirconia, magnesium chloride, montmorillonite, phyllosilicate, zeolites, talc, clays, and the like. Also, combinations of these support materials may be used, for example, silica-chromium, silica-alumina, silica-titania and the like. Additional support materials may include those porous acrylic polymers described in EP 0767184, which is incorporated herein by reference. [0054] The polymer product(s) produced in the reactor can be or include any type of polymer or polymeric material, which in the reactor may constitute at least a portion of the solid particles making up the fluidized bed 112, as noted above; and which may be discharged from the reactor, e.g., via product discharge line 117 as also noted above. For example, the polymer product can include homopolymers of olefins (e.g., homopolymers of ethylene), and/or copolymers, terpolymers, and the like of olefins, particularly ethylene, and at least one other olefin. Illustrative polymers can include, but are not limited to, polyolefins, polyamides, polyesters, polycarbonates, polysulfones, polyacetals, polylactones, acrylonitrile-butadiene-styrene polymers, polyphenylene oxide, polyphenylene sulfide, styrene-acrylonitrile polymers, styrene maleic anhydride, polyimides, aromatic polyketones, or mixtures of two or more of the above. Suitable polyolefins can include, but are not limited to, polymers comprising one or more linear, branched or cyclic C₂ to C40 olefins, preferably polymers comprising propylene copolymerized with one or more C3 to C40 olefins, preferably a C3 to C20 alpha olefin, more preferably C3 to C10 alpha-olefins. More preferred polyolefins include, but are not limited to, polymers comprising ethylene (e.g., polyethylene), including but not limited to ethylene copolymerized with one or more C3 to C40 olefins, preferably one or more C3 to C20 alpha olefins, more preferably C3 to C12 alpha-olefins, such as propylene, 1-butene, 1-hexene, and/or 1-octene. Polymer Products [0055] Preferred polymers include homopolymers or copolymers of C2 to C40 olefins, preferably C₂ to C₂₀ olefins, preferably a copolymer of an alpha-olefin and another olefin or alpha-olefin (ethylene is defined to be an alpha-olefin for purposes of this invention). Preferably, the polymers are or include homo polyethylene, homo polypropylene, propylene copolymerized with ethylene and/or butene, ethylene copolymerized with one or more of propylene, 1-butene, 1-hexene, 1- octene, and optional dienes. Preferred examples include thermoplastic polymers such as ultra low density polyethylene, very low density polyethylene (“VLDPE”), linear low density polyethylene (“LLDPE”), low density polyethylene (“LDPE”), medium density polyethylene (“MDPE”), high density polyethylene (“HDPE”), polypropylene, isotactic polypropylene, highly isotactic polypropylene, syndiotactic polypropylene, random copolymer of propylene and ethylene and/or butene and/or hexene, elastomers such as ethylene propylene rubber, ethylene propylene diene monomer rubber, neoprene, and blends of thermoplastic polymers and elastomers, such as for example, thermoplastic elastomers and rubber toughened plastics. [0056] Polyethylene polymers produced in a gas phase polymerization process are characterized by a number of parameters, including, but not limited to, density, melt index (I2), high load melt index (I21 or HLMI), melt index ratio (MIR), number average molecular weight (Mn), weight average molecular weight (M_w), Z-average molecular weight (M_z), molecular weight distribution (Mw/Mn or MWD), the ratio of the Z-average molecular weight to the weight average molecular weight (Mz/Mw These parameters are related to physical characteristics of polymer chains, including, but not limited to, lengths of polymer chains, distribution of lengths of polymer chains, comonomer distribution among and along polymer chains, and length and number of branches on polymer chains. These physical characteristics of polymer chains lead to different mechanical properties that make different polyethylene polymers suitable for a broad range of end-use applications. [0057] Polymerization conditions in a fluidized bed in a polymerization reaction zone can be controlled both to produce polyethylene polymers having a desired combination of parameters and to maintain the stability of polymerization reaction in a gas phase reactor. Such polymerization conditions include, but are not limited to, reactor temperature, reactor pressure, ethylene monomer feed rate, comonomer type and feed rate, catalyst type and feed rate, comonomer-to-ethylene ratio, rate of addition of hydrogen, an amount of one or more induced condensing agents, an amount of one or more continuity additives, and delta melt initiation temperature (dMIT; see U.S. Pat. No. 7,683,140, the contents of which are fully incorporated by reference herein). [0058] Polyethylene producers typically identify each polyethylene polymer having a particular set of properties by a grade name and/or number. Density and melt index (I₂) are generally key parameters associated with each polyethylene polymer grade. For the producer, each such polyethylene polymer grade is associated with a particular set of polymerization conditions. Continuity Additive/Static Control Agent [0059] In gas-phase polyethylene production processes, it may be desirable to use one or more static control agents to aid in regulating static levels in the reactor. As used herein, a static control agent is a chemical composition which, when introduced into a fluidized bed reactor, may influence or drive the static charge (negatively, positively, or to zero) in the fluidized bed. The specific static control agent used may depend upon the nature of the static charge, and the choice of static control agent may vary dependent upon the polymer being produced and the single site catalyst compounds being used. [0060] Control agents such as aluminum stearate may be employed. The static control agent used may be selected for its ability to receive the static charge in the fluidized bed without adversely affecting productivity. Other suitable static control agents may also include aluminum distearate, ethoxylated amines, and anti-static compositions such as those provided by Innospec Inc. under the trade name OCTASTAT. For example, OCTASTAT 2000 is a mixture of a polysulfone copolymer, a polymeric polyamine, and oil soluble sulfonic acid. [0061] Any of the mentioned control agents may be employed either alone or in combination as a control agent. For example, the carboxylate metal salt may be combined with an amine containing control agent (e.g., a carboxylate metal salt with any family member belonging to the KEMAMINE® (available from Crompton Corporation) or ATMER® (available from ICI Americas Inc.) family of products). [0062] Other useful continuity additives include ethyleneimine additives useful in embodiments disclosed herein may include polyethyleneimines having the following general formula: —(CH2— CH₂—NH)_n-, where n may be from about 10 to about 10,000. The polyethyleneimines may be linear, branched, or hyper branched (e.g., forming dendritic or arborescent polymer structures). They can be a homopolymer or copolymer of ethyleneimine or mixtures thereof (referred to as polyethyleneimine(s) hereafter). Although linear polymers represented by the chemical formula —(CH2—CH2—NH)n- may be used as the polyethyleneimine, materials having primary, secondary, and tertiary branches can also be used. Commercial polyethyleneimine can be a compound having branches of the ethyleneimine polymer. Metallocene Catalyst [0063] In some embodiments, the catalyst used in the first set of polymerization conditions and the second set of polymerization conditions is a metallocene catalyst, which are particularly susceptible to sheeting and/or chunking triggered by changing polymerization conditions. Metallocene catalysts used in the gas phase polymerization process can produce linear low density polyethylene (LLDPE). LLDPEs made using one or more metallocene catalysts are labeled herein as mLLDPE. The mLLDPE can include copolymers of 80 to 99.9 wt% ethylene-derived units, with the balance of units derived from one or more C3 to C12 -olefin comonomers (and in particular one or more of 1-butene, 1-hexene, 1-octene; and more preferably 1-hexene), and may have density ranging from 0.900 g/cm³ to 0.940 g/cm³, such as from a low of any one of 0.905, 0.908, 0.910, 0.912, or 0.915 g/cm³ to a high of any one of 0.920, 0.925, 0.930, 0.935, or 0.940 g/cm³. Metallocene catalysts include, but are not limited to: Type 1: an unbridged bis- cyclopentadienyl Group 4 and substituted versions thereof; Type 2: a bridged bis-cyclopentadienyl Group 4 and substituted versions thereof; Type 3: a substituted bulky ligand hafnium transition metal metallocene-type catalyst compound and substituted versions thereof; and Type 4: a dual catalyst system comprising a bridged bis-cyclopentadienyl Group 4 metal catalyst and an unbridged bis-cyclopentadienyl Group 4 metal catalyst. Examples of suitable metallocene catalyst compounds can include, but are not limited to, those described in U.S. Pat. Nos.: 7,179,876;

[0064] FIG. 1 and many similar illustrations of fluidized beds illustrate a defined “top” of a fluidized bed as if it is the surface of a body of liquid, such that it appears no solid particles are present above this “top” of the bed 112 (just as no liquid particles are present above the top of a body of liquid). However, the reality is not so simple. Solid particles, and in particular fines, will be entrained in the gas flowing up through, and out of, the dense phase of the fluidized bed 112. Thus, for purposes of this application, we refer to the “top” of the fluidized bed 112 as the point above which, with respect to increasing height, solid density continually decreases as compared to a relatively constant solid density level within the fluidized bed (that is, within the “dense phase” of the fluidized bed), until the solid density eventually reaches a relatively constant value (well below the density of the dense phase of the fluidized bed). The solids density in this region can be referred to as the “reflux density” or “expanded section” density. The inventors have observed that the decrease in solids density with respect to increasing height through this region above the “top” of the fluidized bed follows an exponential decay model. [0065] Furthermore, the height above the fluidized bed at which solid density reaches the relatively constant value (even with further increasing height) is known as the “transport disengaging height” (TDH). The TDH is determined by the operating conditions of the fluidized bed (e.g., superficial velocity of gas flowing up through the bed, pressure) as well as the size and shape of the vessel (reactor) in which the fluidized bed is disposed. For example, a cylindrical vessel having a transition section or “neck” with widening diameter, eventually ending in a dome (such as the widening section 105 ending in dome 107 in reactor 101 of FIG.1) results in a region of decreasing gas velocity in the widening-diameter portion, resulting in a lower TDH as compared to a theoretical cylindrical vessel that did not have an increase in diameter. However, on the other hand, the reactor vessel 101 also has a dome 107 above this widening neck section 105, in which diameter again decreases, resulting in a region of increasing gas velocity at the top of the vessel 101. [0066] It is therefore important to control the reactor so that the reflux density (density of fine solid particles refluxing above the fluidized bed) is such that solid particles do not rise above the widest point of the reactor vessel. The space in the reactor vessel 101 between the top of the bed and this widest point (at the transition from neck 105 to dome 107) is referred to as the “freeboard”, and the “freeboard height” is defined as the height between the top of the fluidized bed and said widest point of the vessel 101. This widest point is illustrated, for example, by the dotted line in the reactor vessel profile shown in FIGs.4a and 4b (discussed in more detail below). [0067] In summary, then, it is desired to operate such that the TDH is equal to, or preferably less than, the freeboard height, which means that solid particle density reaches its minimum at a height equal to or less than the freeboard height. Otherwise, if solid particle density is excessive above the freeboard height (meaning solid particles are entrained in gas entering the reactor dome 107), then as superficial gas velocity increases through the dome and into the offtake cycle fluid line 115 and/or vent line 118, the entrained solids are swept along with the fluid and will accumulate in downstream portions of the system, eventually causing fouling and costly reactor shutdowns. [0068] These principles are further illustrated in FIG.2, which is a conceptual graphical depiction (not necessarily to scale) of solids entrainment density (kg/m³ or lb/ft³) (y-axis) vs. height above top of the fluidized bed (Z_f) (x-axis) for an illustrative gas phase fluidized bed system. As shown in FIG. 2, the level of entrainment is highest at the upper part of the fluidized bed, with solids entrainment level in this region represented in FIG.2 as upper fluidized bed bulk density (UFBD) at height Z_f < 0 (where Z_f is height above top of the fluidized bed; ergo height Z_f < 0 corresponds _{to heights within the fluidized bed). In FIG.2, _ is the reflux density at height immediately} above the top of the bed, in the splash zone, taken as Z_f=0), and it can be seen that there is a region of decreasing solid entrainment level with increasing height above bed (increasing Z_f), which corresponds to the reflux or expanded section of the reactor vessel, until the solid entrainment level eventually reaches a minimum value at Zf = TDH. FIG. 2 also illustrates 3 theoretical locations for freeboard height H_f — one undesirably below TDH, one equal to TDH, and one above TDH. As can be seen, where Hf is equal to or above TDH, solids entrainment level is at the minimum at the freeboard height Hf, as is desired. FIG.2 also illustrates both a solid and dotted line for solids entrainment level; the solid line is the solid entrainment trend for the case where H_f = TD or H_f > TDH; while the dotted line is the solid entrainment trend for the case where Hf < TDH. It can be seen that when Hf < TDH, the reflux density is actually lower at a given height Zf as compared to cases when H_f = TDH or H_f > TDH. This is because less solid is refluxing back into the fluidized bed, owing to more solid remaining entrained at the freeboard height (and therefore remaining entrained within the gas flowing up and out of the reactor vessel), and indeed this is illustrated by the higher entrainment density at height Z_f = H_f for the case where H_f < TDH (as shown in FIG. 2) (noting that “entrainment density” is the reflux density at the freeboard height, indicating this is the solids density that will remain entrained in gas exiting the reactor vessel). In contrast, entrainment density is lower (indeed, at the minimum) for the cases where H_f = TDH or H_f > TDH. [0069] With continued reference to FIG. 2, we also see that when H_f > TDH, _{_} (i.e. the density at height Zf = 0) has a value below the UFBD; and when Hf = TDH or Hf < TDH, _{_} is approximately equal to the UFBD, again reflecting that when H_f = TDH or H_f < TDH, a desirably greater amount of solid is refluxing back into the fluidized bed, as illustrated by the higher reflux densities. [0070] In sum, it is desired to find a way to control fluidized bed height accounting for reflux density as accurately as possible, instead of conventional methods of reactor control that rely simply on bed level, or direct detection of solids in the cycle fluid line 115 or other downstream components (at which point, of course, it is too late to control for prevention of solids accumulation). Indeed, accurately detecting solid density at greater heights above the fluidized bed is quite challenging, due to the relatively low absolute values of solid density that are being measured, making it difficult to control the system based upon such actual measurements. [0071] The present inventors have solved this problem by recognizing that accurate measurements can be obtained closer to the fluidized bed, and modeling the density according to the known behavior of the reflux density for given conditions of TDH vs. Hf; and in particular recognizing the exponential nature of the decay of reflux density as height above bed (Z_f) increases up until the TDH. For example, reflux density can be modeled according to the following equation (which conveniently applies to all cases of Hf as compared to TDH – that is, whether Hf is less than, equal to, or greater than, TDH): Equation 1 =_{_}

[0072] In Equation 1, is reflux density at height above the fluidized bed Z_f, _{_} is the reflux density just above the bed (in the splash zone) at Zf=0, and a is an exponential decay constant. Reflux density occurs between Zf > 0 and Zf < Hf. As noted above, when Hf > TDH, then _{Reflux_0} UFBD (upper fluidized bed density). When Hf < TDH, then one has the undesirable condition wherein _{Reflux_0} < UFBD, because in this condition, a greater portion of entrained particles are found higher in the reactor, rather than refluxing back into the bed. The exponential decay constant “a”, which has units of 1/length (e.g., 1/ft or 1/m), can be calculated by plotting the Ln of two or more expanded section densities (that is, density values at two or more heights above fluidized bed Z_f, such as at height Z₁, height Z₂, etc.) against the values of height above the fluidized bed (Zf). The slope of this plotted line is the exponential decay constant “a”. FIG.3 is a natural log versus height graph of three density measurements at three different heights Zf for an illustrative gas phase fluidized bed reactor, illustrating such a plot useful for determination of “a”. [0073] Preferably, three or more densities from each of three or more heights Z_f are measured and plotted against Zf (as just described) to obtain the exponential decay constant “a”; also or instead, one or more densities can be measured from Zf < 0 (that is, within the dense phase of the fluidized bed, such that the measured density used is the UFBD; which also, as discussed above, equals the _ when H_f is equal to or greater than TDH). Similarly, one of the density measurements can be taken at the top of the fluidized bed to obtain _{_} by direct measurement, which could be helpful particularly to guard against cases where TDH becomes greater than the H_f due to operating conditions (because at this point, _{_} might become less than UFBD, and therefore in-bed measurements of density would not necessarily provide a good _{_} value). Alternatively, _{_} can be determined by plotting the Ln of reflux density versus Z_f and determining the y-intercept (at Z_f = 0), which is _{_} , wherein the plot of Ln of reflux density versus Zf is built using the two or more (preferably 3 or more) density measurements each obtained at a height above fluidized bed Zf. [0074] Once the two or more, preferably three or more, density measurements are obtained and the exponential decay constant “a” calculated, the reflux density model (Equation 1) can be used to model reflux density at various heights within the freeboard, up to and including the freeboard height. In particular, reflux density at the freeboard height (that is at height Z_f = freeboard height Hf) is the “entrainment density” (as noted previously). This, of course, is the value which reactor operators would want to minimize so as to minimize solid entrainment in the fluid offtake through cycle fluid line(s) 115 and/or vent line(s) 118 (which, as noted above, can lead to fouling of downstream equipment). So, it is particularly advantageous to use Equation 1 or a similar model to determine entrainment density, using inputs of (i) two or more, preferably three or more, density measurements each obtained at a height above the fluidized bed Z_f, as well as (ii) the corresponding heights Zf at which the density measurements were obtained. [0075] The entrainment density determined from the reflux density model (e.g., Equation 1), in turn, can be used to model entrainment flux, which is the flux of particles (mass flow rate of particles per unit area, typically in lb/ft²-s or kg/m²-s) in the expanded section of the reactor vessel. Entrainment flux models generally have as inputs (1a) a value of entrainment density calculated at Zf = Hf and (1b) a gas velocity at Z_f = H_f within a fluidized bed reactor, with an output of (2) entrainment flux. Equation 2 is an example entrainment flux model. In Equation 2, entrainment flux E can be calculated by multiplying (entrainment density) by the entrainment velocity (Ue) at Zf = Hf, noting that the entrainment velocity is the superficial gas velocity at the widest point in the expanded section of the reactor (that is, at height above the bed Zf = freeboard height H_f). Equation 2 = [0076] Any suitable method or measurement instrument to measure reflux density at a given height (Z_f) above the fluidized bed may be used including, but not limited to, nuclear densometers, and differential pressure instruments, for example. [0077] FIG. 4a is an illustrative depiction of a gas phase fluidized bed reactor (Rx) with 3 differential pressure instruments, labeled D1, D2, and D3, at heights Zf of Z1, Z2, and Z3 above the fluidized bed, for measuring the density of the material within the reactor at heights of Z1, Z2, and Z3, respectively, all disposed below the freeboard height H_f. In FIG. 4a, Z1 is depicted as being at the operating height of the fluidized bed (e.g., such that Zf = Z1 = 0), however, Z1 can be spaced at any height above the fluidized bed. [0078] FIG.4b is an illustrative depiction of a gas phase reactor (Rx) with 3 nuclear densometer instruments, labeled D1, D2, and D3, at heights of Z1, Z2, and Z3 above the fluidized bed, for measuring the density of the material within the reactor at heights of Z1, Z2, and Z3 all disposed below the freeboard height H_f. In FIG. 4b, Z1 is depicted as being at the operating height of the fluidized bed (Zf = Z1 = 0), however, Z1 can be spaced at any height above the fluidized bed. [0079] Although density measurements can be obtained from any of various heights above the bed, it is particularly advantageous, in accordance with some embodiments, to obtain all density measurements within the lower 25% of the freeboard height (that is, such that at least two, preferably at least three, and optionally all, density measurements are obtained at heights above the bed Zf that are equal to 0 to 50%, more preferably 0 to 25%, of Hf). This is because, as noted, measurements at higher points in the freeboard might suffer from reduced accuracy due to the lower solids concentrations, which can lead to exaggerated effects of noise in the data. In this way, a more accurate model of reflux density along the vertical profile of the freeboard may be developed. [0080] Furthermore, a particularly advantageous means of density measurements includes the use of nuclear densometer instruments. Such instruments may include a nuclear radiation source and a corresponding radiation detector configured to detect and quantify the radiation emitted from the nuclear radiation source. The nuclear densometer instruments may be disposed along the outer surface of the fluidized bed reactor between the top end and the bottom end, each at a different vertical distance from the bottom end to irradiate the interior space of the fluidized bed reactor. For instance, a line between each radiation detector and each nuclear radiation source can pass through the interior space of the fluidized bed reactor. The density of material within the fluidized bed reactor can be determined by comparing the intensities of the nuclear radiation measured at the radiation detectors with the corresponding radiation sources. The nuclear densometer instruments can be used to compare the measured intensity of the nuclear radiation at the plurality of radiation detectors, determine the density of material in the reactor at height (Z_f) above the fluidized bed, calculate the exponential decay constant “a”, calculate a reflux density using the exponential decay constant “a” using Equation 1, calculate an entrainment flux using Equation 2, and adjust one or more process variables which are linked to production and/or entertainment based on the entrainment flux. [0081] In some embodiments, the nuclear densometer instruments are located on the outer surface of the reaction zone, the expansion zone (along the outer surface of the reactor neck), or both. In some embodiments, the nuclear densometer instruments include at least 2 nuclear radiation sources or at least 3 nuclear radiation sources, for example, 2 nuclear radiation sources, 3 nuclear radiation sources, 4 nuclear radiation sources, 5 nuclear radiation sources, 6 nuclear radiation sources, 7 nuclear radiation sources, 8 nuclear radiation sources, or any other suitable number of radiation sources. In some embodiments, the nuclear densometer instruments include at least 3 nuclear radiation detectors or at least 4 nuclear radiation detectors, for example 3 nuclear radiation detectors, 4 nuclear radiation detectors, 5 nuclear radiation detectors, 6 nuclear radiation detectors, 7 nuclear radiation detectors, 8 nuclear radiation detectors, 9 nuclear radiation detectors, or any other suitable number of nuclear radiation detectors. [0082] In various embodiments, the density measuring devices including nuclear densometer instruments, and/or differential pressure instruments, can be spaced 2 – 3 ft (0.6 – 0.91 m) apart (in terms of height) along the fluidized bed reactor (e.g., such that Z2 – Z1 in FIG.4a and/or FIG. 4b is within the range of 2 to 3 ft (0.6 to 0.91 m)). In embodiments, a first density measuring device (or a first plurality of density measuring devices intended to measure density at a given height Zf above the fluidized bed) may be placed at or near the top of the upper fluidized bed such as at height Z_f within the range from 0 ft (0m) – 10 ft (3.0 m), or alternatively, at Z_f of about 1 ft (0.3 m), 2 ft (0.6 m), 3 ft (0.9 m), 5 ft (1.52 m), 10 ft (3.0 m), any ranges therebetween, or any other suitable height above the fluidized bed. In some embodiments, the density measuring devices including nuclear densometer instruments, and/or differential pressure instruments, can be spaced apart in increments of 8% to 14% of the freeboard height (Hf) as measured from the top of the fluidized bed. Alternatively, the measuring devices can be spaced apart in increments from 8% to 10% of the freeboard height, 10% to 12% of the freeboard height, 12% to 14% of the freeboard height, or any ranges therebetween. [0083] In some embodiments, the density measuring devices including nuclear densometer instruments, and/or differential pressure instruments, can be integrated into a control system, such as a local control system or a distributed control system. A distributed control system (DCS) is a computer-based control system that is used to monitor and control processes in a chemical plant. [0084] A DCS can be used to control a wide variety of processes, including distillation columns, reactors, and pumps, for example and may be integrated across several units such that simultaneous control of multiple units in response to a single signal may be accomplished. A DCS typically consists of a number of components, including: sensors: these devices measure the physical properties of the process, such as temperature, pressure, and flow rate, controllers: these devices use the data from the sensors to calculate the necessary adjustments to the process, actuators: these devices implement the control commands from the controllers, such as opening or closing valves, and a human-machine interface (HMI): this is the graphical user interface that allows operators to monitor and control the process. The DCS uses various types of logic control such as PID controllers, ladder logic, and sequential function charts to control the processes. The logic control is programmed into the DCS software and is used to ensure that the equipment operates within predefined limits. For example, if a process parameter such as temperature exceeds its setpoint, the DCS will send a signal to a control valve to adjust its position to reduce the temperature. In embodiments, the DCS can take an input signal generated by the density measuring devices and use the input signal to calculate the exponential decay constant “a” in real time and calculate density at any level above the fluidized bed within the fluidized bed reactor. The DCS can also be configured to calculate the entrainment flux and entrainment rate using the calculated density and expanded section gas velocity. [0085] In embodiments, the models disclosed herein can be used to calculate UFBD or _{_} (the reflux density just above the bed, in the splash zone, at Zf=0) and estimate the fluidized bed level. For example, using three radiation sources at Z1=3 ft, Z2=6 ft, and Z3=9 ft above the bed and plotting these three data points on a natural log plot will yield the linear equation with the decay constant as a function of Zf. When Zf=0, then the Y-Intercept is found, which is the _{_} (which, as discussed above, equals UFBD when Hf > TDH).. [0086] In embodiments, the models disclosed herein can be used to calculate solids holdup volume. The solids holdup volume (in units of lbs) is calculated by multiplying the average reflux density (lb/ft3) (for instance, as may be calculated at Zf = Hf/2) by the expanded section volume (ft3). [0087] Once the entrainment rate and/or entrainment flux are calculated, the calculated rate/flux can be compared to a setpoint rate and/or setpoint flux. Thereafter, one or more operating parameters of the fluidized bed reactor may be adjusted such that the entrainment rate and/or entrainment flux are closer to the setpoint rate and/or setpoint flux. For example, a DCS may be configured to adjust the level of solids in the fluidized bed reactor in response to the calculated rate/flux by adjusting one or more process parameters, for example by opening or adjusting one or more valves, compressors, pumps, condensers, boilers, heat exchangers, and the like. [0088] In various embodiments, the process parameters adjusted may be those of the fluidized bed reactor. For example, the DCS may be configured to directly or indirectly adjust one or more of a solids removal rate from the fluidized bed reactor, a fluidization velocity within the fluidized bed reactor, a catalyst feed rate to the fluidized bed reactor, a reactor gas density within the fluidized bed reactor, a reactor gas composition including hydrocarbon molar percent, height of the fluidized bed, a reactor temperature, and/or a reactor pressure. Alternatively to a DCS, the operational parameters of the fluidized bed reactor can be adjusted by a local controller. In further embodiments, when the calculated entrainment rate and/or entrainment flux is greater than a setpoint, a control loop can be initiated which reduces the gas velocity at a top of the reactor, so as to reduce entrainment and avoid fouling due to entrained solids leaving the top of the reactor. On the other hand, when the entrainment rate and/or entrainment flux is below a setpoint, a control loop can be initiated which increases the gas velocity at a top of the reactor, thereby avoiding slow and inefficient operation of the reactor. [0089] In embodiments, if the entrainment rate or flux increases by more than a setpoint such as 25%, 50%, 75%, or 100% more than an established baseline, or is elevated over a defined period of time (e.g.1 hour, 4 hours, 12, hours, 24 hours etc.), then the DCS can generate alarms to alert operators to the reactor condition. Additionally, process parameters can be automatically or manually adjusted to steer the process into a state with less entrainment. In embodiments, some corrective actions include reducing superficial gas velocity through the reactor such as reducing the velocity by 1% to 5%. In further embodiments, the reactor bed level can be decreased by 0.1 ft (0.03 m) to 2 ft (0.6 m). In further embodiments the reactor pressure can be decreased by 1 psig (6.9 kPa) to 10 psig (68.9 kPa). ADDITIONAL EMBODIMENTS [0090] Accordingly, the present disclosure may provide methods directed to gas phase polymerization reactors and, more particularly example embodiments disclose methods of measuring and quantifying entrainment of polymer resin product in a gas phase polymerization reactor. The methods may include any of the various features disclosed herein, including one or more of the following embodiments. [0091] Embodiment 1. A method comprising: obtaining two or more density measurements at two or more heights above a fluidized bed within a fluidized bed reactor; calculating an exponential decay constant using the two or more density measurements at the two or more heights; calculating an entrainment density using a reflux density model which has an input of the exponential decay constant; calculating an entrainment flux using an entrainment flux model which has an input of the entrainment density; and adjusting at least one operating parameter of the fluidized bed reactor in response to the calculated entrainment flux. [0092] Embodiment 2. The method of embodiment 1 wherein the density measurements are obtained by at least one measurement instrument selected from the group consisting of: nuclear densometers, acoustic densometers, optical densometers, diaphragm sensors, differential pressure instruments, and combinations thereof. [0093] Embodiment 3. The method of any of embodiments 1-2 wherein at least one density measurement is a measurement of upper fluidized bed bulk density. [0094] Embodiment 4. The method of any of embodiments 1-3 wherein the two or more density measurements are obtained from locations which differ by about 8% to about 14% of a freeboard height of the fluidized bed reactor. [0095] Embodiment 5. The method of any of embodiments 1-4 wherein the two or more density measurements are obtained from locations which are about 2 ft (0.6 m) to about 3 ft (0.91 m) apart on the fluidized bed reactor. [0096] Embodiment 6. The method of any of embodiments 1-5 wherein the density measurements are obtained by at least two nuclear densometers. [0097] Embodiment 7. The method of any of embodiments 1-6 wherein the reflux density model _{has the form of: = _ where is reflux density, a is the exponential} decay constant, Zf is and _{_} is reflux density above the

fluidized bed at Zf = 0. [0098] Embodiment 8. The method of any of embodiments 1-7 wherein the entrainment flux _{model has the form of: = where E is entrainment, is} entrainment density, U_e is velocity at a widest part of a dome of the fluidized bed reactor. [0099] Embodiment 9. The method of any of embodiments 1-8 wherein adjusting at least one operating parameter of the fluidized bed reactor comprises reducing velocity through the reactor by about 1% to about 5%, decreasing a reactor bed level by about 0.1 ft (0.03 m) to about 2 ft (0.6 m), and/or decreasing pressure in the reactor by about 1 psig (6.9 kPa) to about 10 psig (68.9 kPa). [0100] Embodiment 10. The method of any of embodiments 1-9 wherein the at least one operating parameter of the fluidized bed reactor is selected from the group consisting of fluidized bed height, solids removal rate from the fluidized bed reactor, a fluidization velocity within the fluidized bed reactor, a catalyst feed rate to the fluidized bed reactor, a reactor gas density within the fluidized bed reactor, a feed composition to the fluidized bed reactor, temperature of a feed to the fluidized bed reactor, pressure of a feed to the fluidized bed reactor, height of the fluidized bed reactor, fluidized bed reactor temperature, fluidized bed reactor pressure, and combinations thereof. [0101] Embodiment 11. A method comprising: obtaining two or more density measurements at two or more heights above a fluidized bed within a fluidized bed reactor and inputting the two or more density measurements into a distributed control system (DCS); calculating, using the DCS, an exponential decay constant using the two or more density measurements at the two or more heights, an entrainment density using a reflux density model which has an input of the exponential decay constant, and calculating an entrainment flux using an entrainment flux model which has an input of the entrainment density; comparing, using the DCS, the calculated entrainment flux to a setpoint entrainment flux; and generating a control signal, using the DCS, and sending the control signal to the fluidized bed reactor to adjust at least one operating parameter of the fluidized bed reactor in response to the calculated entrainment flux such that the entrainment flux is closer to the setpoint entrainment flux. [0102] Embodiment 12. The method of embodiment 11 wherein the density measurements are obtained by at least one measurement instrument selected from the group consisting of, nuclear densometers, acoustic densometer, optical densometer, differential pressure instruments, and combinations thereof. [0103] Embodiment 13. The method of any of embodiments 11-12 wherein at least one density measurement is a measurement of upper fluidized bed bulk density. [0104] Embodiment 14. The method of any of embodiments 11-13 wherein the two or more density measurements are obtained from locations which differ by about 8% to about 14% of a freeboard height of the fluidized bed reactor. [0105] Embodiment 15. The method of any of embodiments 11-14 wherein the two or more density measurements are obtained from locations which are about 2 ft (0.6 m) to about 3 ft (0.91 m) apart on the fluidized bed reactor. [0106] Embodiment 16. The method of any of embodiments 11-15 wherein the density measurements are obtained by at least two nuclear densometers. [0107] Embodiment 17. The method of any of embodiments 11-16 wherein the freeboard reflux _{density model has the form of: = _ where is reflux density, a is} the exponential decay constant, Z_f is height above the fluidized bed, and _{_} is the reflux density above the fluidized bed at Zf = 0. [0108] Embodiment 18. The method of any of embodiments 11-17 wherein the entrainment flux _{model has the form of: = where E is entrainment, is} entrainment density, U_e is velocity at a widest part of a dome of the fluidized bed reactor. [0109] Embodiment 19. The method of any of embodiments 11-18 wherein the control signal causes velocity through the reactor to reduce by about 1% to about 5%, wherein the control signal causes a decrease in the fluidized bed by about 0.1 ft (0.03 m) to about 2 ft (0.6 m), and/or wherein the control signal causes a decrease in pressure in the reactor by about 1 psig (6.9 kPa) to about 10 psig (68.9 kPa). [0110] Embodiment 20. The method of any of embodiments 11-19 wherein the at least one operating parameter of the fluidized bed reactor is selected from the group consisting of fluidized bed height, solids removal rate from the fluidized bed reactor, a fluidization velocity within the fluidized bed reactor, a catalyst feed rate to the fluidized bed reactor, a reactor gas density within the fluidized bed reactor, a feed composition to the fluidized bed reactor, temperature of a feed to the fluidized bed reactor, pressure of a feed to the fluidized bed reactor, height of the fluidized bed reactor, fluidized bed reactor temperature, fluidized bed reactor pressure, and combinations thereof. [0111] To facilitate a better understanding of the present invention, the following examples of certain aspects of some embodiments are given. In no way should the following examples be read to limit, or define, the entire scope of the disclosure. [0112] Although the present invention and its advantages have been described in detail, it should be understood that various changes, substitutions and alterations can be made herein without departing from the spirit and scope of the invention as defined by the appended claims. Ranges for various characteristics and attributes disclosed herein are listed as sequentially narrowing ranges. However, it should be understood that any lower endpoint of any ranges can be paired with any upper endpoint for the same characteristic or attribute, and such pairings are also intended to be disclosed herein. All patents, test procedures, and other documents cited in this application are fully incorporated herein by reference for all jurisdictions in which such incorporation is permitted. Moreover, the scope of the present application is not intended to be limited to the particular embodiments of the processes, machines, means, methods, and/or steps described in the specification. As one of the ordinary skill in the art will readily appreciate from the disclosure of the present invention, processes, machines, means, methods, and/or steps, presently existing or later to be developed that perform substantially the same function or achieve substantially the same result as the corresponding embodiments described herein, may be utilized according to the present invention. Accordingly, the appended claims are intended to include within their scope such processes, machines, means, methods, and/or steps.

Claims

CLAIMS What is claimed is: 1. A method comprising: obtaining two or more density measurements at two or more heights above a fluidized bed within a fluidized bed reactor; calculating an exponential decay constant using the two or more density measurements at the two or more heights; calculating an entrainment density using a reflux density model which has an input of the exponential decay constant; calculating an entrainment flux using an entrainment flux model which has an input of the entrainment density; and adjusting at least one operating parameter of the fluidized bed reactor in response to the calculated entrainment flux.

2. The method of claim 1 wherein the density measurements are obtained by at least one measurement instrument selected from the group consisting of: nuclear densometers, acoustic densometers, optical densometers, diaphragm sensors, differential pressure instruments, and combinations thereof.

3. The method of claim 2, wherein the density measurements are obtained by nuclear densometers.

4. The method of claim 1 or any one of claims 2-3, further comprising measuring an upper fluidized bed bulk density.

5. The method of claim 1 or any one of claims 2-4, wherein the two or more density measurements are obtained from heights within 0% to about 25% of a freeboard height in the fluidized bed reactor. 6. The method of claim 1 or any one of claims 2-5, wherein the two or more density measurements are obtained from heights which are about 2 ft (0.

6 m) to about 3 ft (0.91 m) apart in the fluidized bed reactor.

7. The method of claim 1 or any one of claims 2-6, wherein the reflux density model has the form of: =_{_}

where is reflux a constant, Zf is height above the fluidized bed, and _{_} is reflux density immediately above the top of the fluidized bed at Z_f = 0.

8. The method of claim 1 or any one of claims 2-7, wherein the entrainment flux model has the form of: = where E is entrainment, is entrainment density, and Ue is superficial gas velocity at a widest part of the fluidized bed reactor.

9. The method of claim 1 or any one of claims 2-8, wherein adjusting at least one operating parameter of the fluidized bed reactor comprises reducing superficial gas velocity through the reactor, decreasing a reactor bed level, and/or decreasing pressure in the reactor.

10. The method of claim 1 or any one of claims 2-9, wherein the at least one operating parameter of the fluidized bed reactor is selected from the group consisting of fluidized bed height, solids removal rate from the fluidized bed reactor, superficial gas velocity within the fluidized bed reactor, a catalyst feed rate to the fluidized bed reactor, a reactor gas density within the fluidized bed reactor, a feed composition to the fluidized bed reactor, temperature of a feed to the fluidized bed reactor, pressure of a feed to the fluidized bed reactor, height of the fluidized bed reactor, fluidized bed reactor temperature, fluidized bed reactor pressure, and combinations thereof.

11. A method comprising: obtaining two or more density measurements at two or more heights above a fluidized bed within a fluidized bed reactor and inputting the two or more density measurements into a distributed control system (DCS); calculating, using the DCS, an exponential decay constant using the two or more density measurements at the two or more heights, an entrainment density using a reflux density model which has an input of the exponential decay constant, and calculating an entrainment flux using an entrainment flux model which has an input of the entrainment density; comparing, using the DCS, the calculated entrainment flux to a setpoint entrainment flux; and generating a control signal, using the DCS, and sending the control signal to the fluidized bed reactor to adjust at least one operating parameter of the fluidized bed reactor in response to the calculated entrainment flux such that the entrainment flux becomes closer to the setpoint entrainment flux.

12. The method of claim 11 wherein the density measurements are obtained by at least one measurement instrument selected from the group consisting of nuclear densometers, acoustic densometer, optical densometer, differential pressure instruments, and combinations thereof.

13. The method of claim 12, wherein the density measurements are obtained by nuclear densometers.

14. The method of claim 11 or any one of claims 12-13, wherein at least one density measurement is a measurement of upper fluidized bed bulk density.

15. The method of claim 11 or any one of claims 12-14, wherein the two or more density measurements are obtained from heights which are within 0% to about 25% of a freeboard height in the fluidized bed reactor.

16. The method of claim 11 or any one of claims 12-15, wherein the two or more density measurements are obtained from heights which are about 2 ft (0.6 m) to about 3 ft (0.91 m) apart in the fluidized bed reactor.

17. The method of claim 11 or any one of claims 12-16, wherein the freeboard reflux density model has the form of: _{= _} where is reflux

constant, Zf is height above the fluidized bed, and _{_} is the reflux density above the fluidized bed at Zf = 0.

18. The method of claim 11 or any one of claims 12-17, wherein the entrainment flux model has the form of: = where E is entrainment, Ue is superficial gas velocity at a

widest part of a dome of the reactor.

19. The method of claim 11 or any one of claims 12-18, wherein the control signal causes superficial gas velocity through the reactor to reduce by about 1% to about 5%, wherein the control signal causes a decrease in the fluidized bed by about 0.1 ft (0.03 m) to about 2 ft (0.6 m), and/or wherein the control signal causes a decrease in pressure in the reactor by about 1 psig (6.9 kPa) to about 10 psig (68.9 kPa).

20. The method of claim 11 or any one of claims 12-19, wherein the at least one operating parameter of the fluidized bed reactor is selected from the group consisting of fluidized bed height, solids removal rate from the fluidized bed reactor, a superficial gas velocity within the fluidized bed reactor, a catalyst feed rate to the fluidized bed reactor, a reactor gas density within the fluidized bed reactor, a feed composition to the fluidized bed reactor, temperature of a feed to the fluidized bed reactor, pressure of a feed to the fluidized bed reactor, height of the fluidized bed reactor, fluidized bed reactor temperature, fluidized bed reactor pressure, and combinations thereof.