A gas-liquid two-phase flow measuring system and a measuring method

Abstract

The application discloses a kind of gas-liquid two-phase flow measurement system and measurement method described ultrasonic flowmeter and described positive displacement flowmeter are sequentially arranged in sequence on the first pipeline;Described ultrasonic flowmeter is equipped with pressure measurement unit;Temperature measurement unit is equipped between described ultrasonic flowmeter and described positive displacement flowmeter;Described ultrasonic flowmeter is used to measure the flow value of fluid passing in a certain time;Described positive displacement flowmeter is used to measure the flow value of fluid passing in a certain time.The beneficial effects of the application are that by simultaneously increasing ultrasonic flowmeter and positive displacement flowmeter in the first pipeline, the flow of gas-liquid two-phase flow in the pipeline is measured respectively, and the gas phase flow value and the liquid phase flow value are calculated through the calculation relationship of the fluid in the pipeline, realizing the process of real-time online detection of gas phase flow value and liquid phase flow value without separating gas-liquid two-phase flow.

Description

Technical Field

[0001] The present invention relates to the technical field of gas-liquid two-phase flow measurement, and in particular to a gas-liquid two-phase flow measurement system and a measurement method. Background Art

[0002] Gas-liquid two-phase flow refers to the process of fluid flow in a pipeline with two different phases, namely gas and liquid. Since both gas and liquid phases are fluids, there is a deformable interface between the two phases. The gas phase is a compressible fluid and the liquid phase is an incompressible fluid. There are significant differences in the physical and chemical properties of the gas and liquid phases, and there is velocity slip between the gas and liquid phases, so the flow process is very complex. There are many studies on this type of flow at home and abroad, but due to the difficulty, it has not developed rapidly. Gas-liquid two-phase flow is commonly found in the production process of the upstream field of the oil and gas industry. For example, the natural gas produced at the natural gas wellhead is wet natural gas containing water. In order to meet the application requirements of industrial field production, it is necessary to accurately measure the gas-liquid mixed fluid produced at the wellhead.

[0003] At present, the main method for measuring wet gas and gas-liquid two-phase flow in the oil and gas industry is separation. That is, a separator is used to separate the gas-liquid two-phase flow into gas phase and liquid phase, and then single-phase instruments are used to measure them respectively. The equipment is large, the process flow is complex, the cost is high, and it requires dedicated maintenance. Most of the time, the measurement is time-sharing and rotation, and continuous online measurement cannot be achieved.

[0004] In view of this, this application is hereby filed. Summary of the Invention

[0005] The technical problem to be solved by the present invention is that in the existing technology, when measuring gas-liquid two-phase flow, the gas-liquid two-phase flow is measured separately, and it is impossible to achieve continuous online measurement of the volume of the transmitted fluid. The purpose is to provide a gas-liquid two-phase flow measurement system and measurement method, which realizes direct online real-time measurement of the volume of the flowing fluid.

[0006] The present invention is achieved through the following technical solutions:

[0007] A gas-liquid two-phase flow measurement system includes a first pipeline, an ultrasonic flowmeter, and a positive displacement flowmeter, wherein the ultrasonic flowmeter and the positive displacement flowmeter are sequentially arranged on the first pipeline; a pressure measuring unit and a temperature measuring unit are further provided between the ultrasonic flowmeter and the positive displacement flowmeter, and the pressure measuring unit and the temperature measuring unit are sequentially arranged on the first pipeline;

[0008] The ultrasonic flowmeter is used to measure the flow rate of the fluid passing through within a certain period of time;

[0009] The volumetric flow meter is used to measure the flow rate of a fluid passing through within a certain period of time.

[0010] Traditionally, when measuring the fluid flow of a gas-liquid two-phase flow, a separation method is often used to separate the gas phase fluid from the liquid phase fluid, and then use single-phase instruments to measure the obtained gas phase fluid and liquid phase fluid respectively. However, when using this method to measure the gas-liquid two-phase flow, the equipment used is relatively large and requires operators to maintain the equipment. Moreover, when measuring the fluid, it can only be achieved by time-sharing rotation metering, and it is impossible to perform online real-time metering of the output fluid flow. The present invention provides a gas-liquid two-phase flow measurement system, which measures the flow of the gas-liquid two-phase flow in the pipeline by adding an ultrasonic flowmeter and a volumetric flowmeter at the same time in a first pipeline, and calculates the gas phase flow rate value and the liquid phase flow rate value through the calculation relationship of the fluid in the pipeline, thereby realizing the process of real-time online detection of the gas phase flow rate value and the liquid phase flow rate value when the gas-liquid two-phase flow is not separated.

[0011] Preferably, the system further comprises a desander and a filter, the output end of the desander is connected to the input end of the filter, and the output end of the filter is connected to the first pipeline.

[0012] Adding a desander and a filter to the system is used to filter out impurities from the output gas, thereby increasing the accuracy of the calculation of the gas-liquid two-phase flow rate.

[0013] Preferably, the ultrasonic flowmeter is embedded in the first pipe, and the length range of the ultrasonic flowmeter embedded in the first pipe is: d is the diameter length of the first pipe.

[0014] Preferably, the ultrasonic flowmeter is a through-beam ultrasonic flowmeter, and the through-beam ultrasonic flowmeter is installed with a vertically upward flow direction or a vertically downward flow direction.

[0015] Preferably, the volumetric flowmeter is a Roots flowmeter, a scraper flowmeter, or a reciprocating piston flowmeter.

[0016] Preferably, the temperature measuring unit is a temperature sensor, and the pressure measuring unit is a pressure sensor.

[0017] The present invention also provides a gas-liquid two-phase flow measurement method, which uses the above-mentioned measurement system for measurement, and the method steps include:

[0018] S1: measuring the fluid flow rate value V1 flowing through a certain period of time by the volumetric flow meter;

[0019] S2: measuring the flow rate value V2 of the fluid flowing through within a certain period of time by the ultrasonic flow meter;

[0020] S3: measuring the temperature value in the first pipeline by the temperature measuring unit and the pressure value in the first pipeline by the pressure measuring unit, and calculating the slip ratio s of the gas-liquid flow rate based on the parameter temperature value and the pressure value;

[0021] S4: Based on the parameters V1, V2, and s, calculate the gas phase fluid flow value V a and the liquid fluid flow value V g .

[0022] Preferably, the sub-steps of step S3 include:

[0023] measuring a temperature value in the first pipe by the temperature measuring unit, and measuring a pressure value in the first pipe by the pressure measuring unit;

[0024] Based on the parameter temperature and pressure values, the gas state equation is used to calculate the gas phase fluid density ρ g , using a liquid density meter to measure the liquid phase fluid density ρ l ;;

[0025] Based on the parameter ρ g , ρ l , calculate and obtain the slip speed ratio s.

[0026] Preferably, the liquid phase fluid flow value V g Specific expression and gas phase fluid flow rate value V a The specific expressions are:

[0027]

[0028]

[0029] Preferably, the slip ratio s is specifically expressed as:

[0030]

[0031] α is a fixed coefficient.

[0032] Compared with the prior art, the present invention has the following advantages and beneficial effects:

[0033] The present invention provides a gas-liquid two-phase flow measurement system and method. This system uses a positive displacement flowmeter to measure the total volume of the gas-liquid two-phase flow, an ultrasonic flowmeter to measure the gas phase flow velocity and apparent flow rate in the gas-liquid two-phase flow, and a temperature and pressure measurement unit to measure the temperature and pressure of the gas-liquid two-phase fluid, thereby achieving density measurement of the gas phase fluid in the gas-liquid two-phase flow. By analyzing the gas and liquid densities in the gas-liquid two-phase flow and calculating the flow velocity difference between the gas and liquid, the measurement results of the positive displacement flowmeter and the ultrasonic flowmeter are combined to achieve measurement of the gas and liquid phase flow rates. BRIEF DESCRIPTION OF THE DRAWINGS

[0034] In order to more clearly illustrate the technical solutions of the exemplary embodiments of the present invention, the following briefly introduces the drawings required for use in the embodiments. It should be understood that the following drawings only illustrate certain embodiments of the present invention and therefore should not be regarded as limiting the scope. For ordinary technicians in this field, other relevant drawings can be obtained based on these drawings without making any creative efforts.

[0035] Figure 1 Schematic diagram of the measurement system

[0036] Figure 2 Schematic diagram of ultrasonic flowmeter

[0037] Figure 3 Schematic diagram of the measurement method

[0038] Figure 4 Detailed diagram for two flowmeter settings DETAILED DESCRIPTION

[0039] In order to make the objectives, technical solutions and advantages of the present invention more clearly understood, the present invention is further described in detail below in conjunction with examples and drawings. The exemplary embodiments of the present invention and their descriptions are only used to explain the present invention and are not intended to limit the present invention.

[0040] In the following description, numerous specific details are set forth to provide a thorough understanding of the present invention. However, it will be apparent to one skilled in the art that these specific details are not necessarily required to practice the present invention. In other embodiments, well-known structures, circuits, materials, or methods are not described in detail to avoid obscuring the present invention.

[0041] Throughout this specification, references to "one embodiment," "an embodiment," "an example," or "an example" mean that a particular feature, structure, or characteristic described in connection with the embodiment or example is included in at least one embodiment of the present invention. Therefore, appearances of the phrases "one embodiment," "an embodiment," "an example," or "an example" in various places throughout this specification are not necessarily all referring to the same embodiment or example. Furthermore, the particular features, structures, or characteristics may be combined in one or more embodiments or examples in any suitable combinations and / or subcombinations. Furthermore, it will be understood by those of ordinary skill in the art that the figures provided herein are for illustrative purposes only and are not necessarily drawn to scale. As used herein, the term "and / or" includes any and all combinations of one or more of the associated listed items.

[0042] In the description of the present invention, the terms "front", "back", "left", "right", "up", "down", "vertical", "horizontal", "high", "low", "inside", "outside", etc., indicating directions or positional relationships, are based on the directions or positional relationships shown in the accompanying drawings. They are only for the convenience of describing the present invention and simplifying the description, and do not indicate or imply that the device or element referred to must have a specific direction, be constructed and operated in a specific direction. Therefore, they should not be understood as limiting the scope of protection of the present invention.

[0043] Example 1

[0044] This embodiment provides a gas-liquid two-phase flow measurement system. In the technical solution provided by this embodiment, there is no need to separate the wet natural gas two-phase flow. The gas and liquid phase flow rates in the gas-liquid two-phase flow mixture are calculated based on the analysis and derivation of the measurement results of the ultrasonic flowmeter and the volumetric flowmeter 4. The total volume of the gas-liquid two-phase flow is measured using the volumetric flowmeter 4, the gas phase flow velocity and apparent flow rate in the gas-liquid two-phase flow are measured using the ultrasonic flowmeter, and the temperature and pressure measurement unit 6 is used to measure the temperature and pressure of the gas-liquid two-phase fluid to achieve density measurement of the gas phase fluid in the gas-liquid two-phase flow. By analyzing the gas density and liquid density in the gas-liquid two-phase flow, the flow velocity difference between the gas and liquid is calculated. The gas phase flow rate and liquid phase flow rate are measured by combining the measurement results of the volumetric flowmeter 4 and the ultrasonic flowmeter.

[0045] like Figure 1 and Figure 4As shown, the measurement system includes a first pipeline 7, an ultrasonic flowmeter 3 and a volumetric flowmeter 4, and the ultrasonic flowmeter 3 and the volumetric flowmeter 4 are sequentially arranged on the first pipeline 7; a pressure measuring unit 6 is provided on the ultrasonic flowmeter 3; a temperature measuring unit 5 is provided between the ultrasonic flowmeter 3 and the volumetric flowmeter 4; the ultrasonic flowmeter 3 is used to measure the flow value of the fluid passing through within a certain period of time; the volumetric flowmeter 4 is used to measure the flow value of the fluid passing through within a certain period of time.

[0046] The ultrasonic flowmeter 3 and the volumetric flowmeter 4 are arranged in the first pipe 7. The ultrasonic flowmeter 3 can be used to measure the flow value of the fluid flowing through the ultrasonic flowmeter 3, and the volumetric flowmeter 4 can be used to measure the flow value of the fluid flowing through the volumetric flowmeter 4, and calculations can be performed based on the obtained flow values.

[0047] In this embodiment, a pressure measuring unit 6 is provided in the first pipe 7, and a pressure sensor is used to detect the pressure value in the first pipe 7. The temperature measuring unit 5 is set as a temperature sensor to measure the real-time temperature in the first pipe 7. When the operator obtains the pressure and temperature values ​​through real-time measurement, the operator can use the gas state equation and the liquid state method to calculate the corresponding gas phase fluid density ρ. g and the density of the liquid phase ρ l , and calculate the slip ratio s of the gas-liquid two-phase flow based on the obtained density.

[0048] In this embodiment, before the ultrasonic flowmeter 3 is set, the system also includes a desander 1 and a filter 2. The output end of the desander 1 is connected to the input end of the filter 2, and the output end of the filter 2 is connected to the first pipe 7. In industry, the gas-liquid two-phase flow directly obtained contains certain solid impurities. Therefore, in order to make the flow value of the gas-liquid two-phase flow more accurately calculated, it is necessary to filter out these fixed impurities, so that the final calculated result can be more accurate.

[0049] The ultrasonic flowmeter is embedded in the first pipe, and the length range of the ultrasonic flowmeter embedded in the first pipe is: d is the diameter of the first pipe. The ultrasonic flowmeter is embedded in the inner wall of the first pipe at a certain distance, so that the flow in the first pipe can be measured.

[0050] In this embodiment, the ultrasonic flowmeter 3 is set as a through-type ultrasonic flowmeter, but this embodiment does not impose any specific restrictions on the specific setting of the ultrasonic flowmeter 3, and the through-type ultrasonic flowmeter is installed with a vertical upward flow direction or a vertical downward flow direction.

[0051] In this embodiment, the volumetric flowmeter 4 is a Roots flowmeter, a scraper flowmeter, or a reciprocating piston flowmeter, but this embodiment does not limit the specific arrangement of the flowmeter.

[0052] Specific measurement principle:

[0053] The measurement principle of the positive displacement flowmeter 4 is to use a fixed displacement method, using mechanical measuring elements to continuously divide the fluid into individual known volumes. This process is repeated, with each volume filled and drained, to cumulatively measure the total fluid flow rate. Therefore, the positive displacement flowmeter 4 can accurately measure the total volume of both the gas and liquid phases.

[0054] When the gas-liquid two-phase flow passes through a positive displacement flowmeter, equation (1) can be obtained, where V a is the volume flow rate of the liquid phase in the gas-liquid two-phase fluid, V g is the volume flow rate of the gas phase in the gas-liquid two-phase flow, V1 is the volume flow rate measured by the positive displacement flowmeter

[0055] V a +V g =V1(1)

[0056] When the gas-liquid two-phase flow flows through the ultrasonic flowmeter, as shown in the figure below, the gas-liquid two-phase flow presents an annular flow state in the vertical pipe, that is, the gas flows in the center of the pipe and the liquid flows around the pipe. The flow rate of the gas is u g , the gas flow area is A g , the flow rate of the liquid is u a , the flow area of ​​the liquid is A a The flow area of ​​the pipe is A. According to the relationship between flow rate and flow velocity, there is the following relationship:

[0057] u g *A g =V g (2)

[0058] u a *A a =V a (3)

[0059] A a +A g =A (4)

[0060] like Figure 2As shown in the figure, the basic measurement principle of the ultrasonic flowmeter is to measure the flow velocity of the fluid by using the difference in the propagation velocity of ultrasound in the forward and reverse strokes in the pipeline, and to obtain the flow rate of the fluid in the pipeline by using the measured flow velocity and the flow area of ​​the pipeline. When the ultrasonic flowmeter is used for gas-liquid two-phase flow measurement, it can still accurately measure the flow velocity of the gas phase fluid in the gas-liquid two-phase flow. At the same time, for the ultrasonic flowmeter, its pipeline flow area is fixed, that is, the pipeline flow area A. At this time, the volume measurement value V2 of the ultrasonic flowmeter has the following relationship;

[0061] V2=u g *A (5)

[0062] Substituting formula (4) into formula (5), we can get

[0063] V2=u g *(A g +A a )=u g *A g +u g *A a (6)

[0064] In the study of gas-liquid two-phase flow, there is a concept of gas-liquid flow rate slip ratio s, which is defined as the ratio of gas phase flow rate to liquid phase flow rate, and is defined as follows (7):

[0065] s=u g / u a (7)

[0066] According to the definition of gas-liquid slip ratio (7), we can get

[0067] u g =u a *s (8)

[0068] Substituting (8) into (6), we can get

[0069] V2=u g *(A g +A a )=u g *A g +s*u a *A a (9)

[0070] Substituting the relationship between (3) and (4) into (9), we can get

[0071] V2=u g *(A g +A a )=u g *A g +s*u a*A a =V g +s*V a (10)

[0072] Using the two relationships (10) and (1), the gas flow rate V can be obtained: g and liquid fluid V l They are shown as follows

[0073]

[0074]

[0075] According to research, the gas-liquid two-phase flow rate sliding ratio s is mainly related to the density of the gas phase and the liquid phase. Within the typical measurement range, there is the following approximate relationship:

[0076]

[0077] α is an undetermined coefficient that needs to be determined through experiments; ρ g is the gas phase fluid density, ρ l is the density of the liquid phase fluid. For this measurement system, the liquid density is a known quantity, measured offline through regular sampling and then provided as a parameter to the system. Liquid density is typically measured using a liquid densitometer.

[0078] V1 is the volume flow measurement value of the positive displacement flowmeter, V2 is the volume flow measurement value of the ultrasonic flowmeter, P is the pressure measurement value of the pressure unit, T is the temperature measurement value of the temperature measurement unit, ρ g is the gas phase fluid density in gas-liquid two-phase flow, ρ l is the density of the liquid phase in the gas-liquid two-phase flow, u g is the gas phase fluid velocity in the pipeline, u l is the flow rate of the liquid phase in the pipeline.

[0079] Density of gas phase fluid ρ g Using the gas phase composition information and the temperature and pressure measured by the pressure and temperature measurement unit, the gas state equation is used for calculation. The density of the liquid phase fluid is calculated by ρ l Therefore, by determining and establishing the functional relationship between the gas-liquid slip ratio and the gas-liquid density ratio through preliminary experiments, the flow rate of the gas and liquid phases in the gas-liquid two-phase flow can be measured by combining the positive displacement flowmeter 4 with the ultrasonic flowmeter.

[0080] This embodiment discloses a gas-liquid two-phase flow measurement system that uses a positive displacement flowmeter 4 to measure the total volume of the gas-liquid two-phase flow, an ultrasonic flowmeter to measure the gas phase flow velocity and apparent flow rate in the gas-liquid two-phase flow, and a temperature and pressure measurement unit 6 to measure the temperature and pressure of the gas-liquid two-phase fluid, thereby measuring the density of the gas phase in the gas-liquid two-phase flow. By analyzing the gas and liquid densities in the gas-liquid two-phase flow, the flow velocity difference between the gas and liquid is calculated, and the measurement results of the positive displacement flowmeter 4 and the ultrasonic flowmeter are combined to achieve measurement of the gas and liquid phase flow rates.

[0081] Example 2

[0082] This embodiment discloses a gas-liquid two-phase flow measurement method. Figure 3 As shown, this embodiment uses the measurement system in the first embodiment for measurement. In this embodiment, the calculation principle of the method is the same as that of the first embodiment, and the method steps include:

[0083] S1: measuring the fluid flow rate value V1 flowing through a certain period of time by the volumetric flow meter;

[0084] S2: measuring the flow rate value V2 of the fluid flowing through within a certain period of time by the ultrasonic flow meter;

[0085] S3: measuring the temperature value in the first pipeline by the temperature measuring unit and the pressure value in the first pipeline by the pressure measuring unit, and calculating the slip ratio s of the gas-liquid flow rate based on the parameter temperature value and the pressure value;

[0086] The sub-steps of step S3 include:

[0087] measuring a temperature value in the first pipe by the temperature measuring unit, and measuring a pressure value in the first pipe by the pressure measuring unit;

[0088] Based on the parameter temperature and pressure values, the gas state equation is used to calculate the gas phase fluid density ρ g and the liquid phase density ρ l ;

[0089] Based on the parameter ρ g , ρ l , calculate and obtain the slip speed ratio s.

[0090] S4: Based on the parameters V1, V2, and s, calculate the gas phase fluid flow value V a and the liquid fluid flow value V g .

[0091] The liquid phase fluid flow value V g Specific expression and gas phase fluid flow rate value Va The specific expressions are:

[0092]

[0093]

[0094] The specific expression of the slip ratio s is:

[0095]

[0096] α is a fixed coefficient.

[0097] V1 is the volume flow measurement value of the positive displacement flowmeter, V2 is the volume flow measurement value of the ultrasonic flowmeter, P is the pressure measurement value of the pressure unit, T is the temperature measurement value of the temperature measurement unit, ρ g is the gas phase fluid density in gas-liquid two-phase flow, ρ l is the density of the liquid phase in the gas-liquid two-phase flow, u g is the gas phase fluid velocity in the pipeline, u l is the flow rate of the liquid phase in the pipeline.

[0098] The specific implementation methods described above further illustrate the objectives, technical solutions and beneficial effects of the present invention in detail. It should be understood that the above description is only a specific implementation method of the present invention and is not intended to limit the scope of protection of the present invention. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principles of the present invention should be included in the scope of protection of the present invention.

Claims

1. A gas-liquid two-phase flow measurement system, characterized in that: The invention comprises a first pipeline (7), an ultrasonic flowmeter (3) and a positive displacement flowmeter (4), wherein the ultrasonic flowmeter (3) and the positive displacement flowmeter (4) are sequentially arranged on the first pipeline (7); a pressure measuring unit (6) and a temperature measuring unit (5) are further arranged between the ultrasonic flowmeter (3) and the positive displacement flowmeter (4), and the pressure measuring unit (6) and the temperature measuring unit (5) are sequentially arranged on the first pipeline (7); The ultrasonic flowmeter (3) is used to measure the flow value of a fluid passing through within a certain period of time; The volumetric flow meter (4) is used to measure the flow value of a fluid passing through within a certain period of time.

2. A gas-liquid two-phase flow measurement system according to claim 1, characterized in that: The system further comprises a desander (1) and a filter (2), wherein the output end of the desander (1) is connected to the input end of the filter (2), and the output end of the filter (2) is connected to the first pipeline (7).

3. A gas-liquid two-phase flow measurement system according to claim 2, characterized in that: The ultrasonic flowmeter (3) is embedded in the first pipe (7) and is provided, and the length range of the ultrasonic flowmeter (3) embedded in the first pipe (7) is: , d is the diameter length of the first pipe (7).

4. A gas-liquid two-phase flow measurement system according to claim 2, characterized in that: The ultrasonic flowmeter (3) is a beam-type ultrasonic flowmeter, and the beam-type ultrasonic flowmeter is installed in a vertically upward flow direction or a vertically downward flow direction.

5. A gas-liquid two-phase flow measurement system according to claim 4, characterized in that: The volumetric flowmeter (4) is a Roots flowmeter, a scraper flowmeter, or a reciprocating piston flowmeter.

6. A gas-liquid two-phase flow measurement system according to claim 4, characterized in that: The temperature measuring unit (5) is a temperature sensor, and the pressure measuring unit (6) is a pressure sensor.

7. A gas-liquid two-phase flow measurement method, characterized in that: The measurement system according to any one of claims 1 to 6 is used for measurement, and the method steps include: S1: The volumetric flow meter (4) measures the flow rate of the fluid flowing through the device within a certain period of time. ; S2: The ultrasonic flowmeter (3) measures the flow rate of the fluid flowing through the device within a certain period of time. ; S3: measuring the temperature value in the first pipe (7) by the temperature measuring unit (5), and measuring the pressure value in the first pipe (7) by the pressure measuring unit (6), and calculating the slip ratio s of the gas-liquid flow rate based on the parameter temperature value and the pressure value; S4: Parameter-based 、 , s, calculate the gas phase fluid flow value Liquid phase fluid flow value .

8. A gas-liquid two-phase flow measurement method according to claim 7, characterized in that: The sub-steps of step S3 include: Measuring a temperature value in the first pipe (7) by the temperature measuring unit (5), and measuring a pressure value in the first pipe (7) by the pressure measuring unit (6); Based on the parameter temperature and pressure values, the gas state equation is used to calculate the gas phase fluid density. , using a liquid density meter to measure the density of liquid fluid ; Parameter-based 、 , calculate and obtain the slip speed ratio s.

9. A gas-liquid two-phase flow measurement method according to claim 8, characterized in that: The liquid phase fluid flow value Specific expressions and gas phase fluid flow values The specific expressions are:

10. A gas-liquid two-phase flow measurement method according to claim 8, characterized in that: The specific expression of the slip ratio s is: α is a fixed coefficient.

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