An electric precipitator inlet flow uniformization system and method

Abstract

This disclosure provides an inlet flow equalization system and method for an electrostatic precipitator, belonging to the field of electrostatic precipitator technology. The system includes a self-adjusting flow equalization plate assembly and a pre-charging device. The self-adjusting flow equalization plate assembly includes a flow equalization plate body, an adjustment mechanism, and a control unit. The control unit calculates the target flow equalization plate angle based on the flue gas flow rate and velocity, and adaptively adjusts the angle of the flow equalization plate body through the adjustment mechanism. The pre-charging device includes a pre-charging component disposed in front of the airflow distribution plate and an electromagnetic hammer rapping unit. The cathode frame of the pre-charging component forms an anode-cathode connection with the airflow distribution plate to achieve dust pre-charging and collection. The electromagnetic hammer rapping unit is used for timed dust removal. This disclosure achieves automatic and precise adjustment of the flow equalization plate angle through the self-adjusting flow equalization plate assembly, improving the problem of uneven inlet flue gas distribution; and achieves dust pre-charging and partial dust collection through the pre-charging device, improving overall dust removal efficiency.

Description

Technical Field

[0001] The embodiments disclosed herein belong to the field of electrostatic precipitator technology, specifically relating to an inlet flow equalization system and method for an electrostatic precipitator. Background Technology

[0002] In modern industrial production, electrostatic precipitators (ESPs), as core equipment for dust control, are widely used in industries such as power, steel, and cement due to their high dust removal efficiency, playing a significant role in ensuring a clean production environment and reducing air pollution. However, ESPs currently face the following common problems during operation: 1. Uneven distribution of flue gas at the inlet. Uneven distribution of flue gas at the inlet of the electrostatic precipitator and the tendency for airflow to generate vortices cause disorder in the internal electric field distribution of the electrostatic precipitator, resulting in a significant reduction in dust charging and collection efficiency, which severely weakens the overall dust removal effect of the electrostatic precipitator.

[0003] 2. To alleviate the above problems, installing inlet flow equalization plates has become a common practice. However, existing inlet flow equalization plates have complex structural designs, requiring significant manpower, resources, and time for actual installation and commissioning, and are extremely difficult to adjust. More importantly, existing flow equalization plates cannot adaptively adjust to the actual operating conditions of the electrostatic precipitator, such as large fluctuations in flue gas flow and unstable changes in flow velocity, making it difficult to maintain good flow equalization in complex and variable working environments.

[0004] 3. Lack of pre-charging function: Conventional airflow distribution plates can only achieve preliminary uniform flow of flue gas and cannot capture dust. During the dust charging process, the high wind speed in the electric field leads to poor dust collection effect of the anode plates at the front of the electric field, resulting in wasted dust collection area and further weakening the dust removal efficiency. Existing technology simply arranges conventional airflow distribution plates at the inlet horn, which cannot achieve pre-charging of dust and cannot fundamentally improve the performance of the electrostatic precipitator.

[0005] Therefore, there is an urgent need to develop an electrostatic precipitator inlet device that can achieve self-regulating flow equalization, has a pre-charging function, has a reasonable structure, and is easy to adjust. Summary of the Invention

[0006] The embodiments disclosed herein aim to at least solve one of the problems existing in the inlet devices of electrostatic precipitators, such as poor flow equalization effect, inability to precharge, complex structure and inconvenient adjustment, and provide an electrostatic precipitator inlet flow equalization system and method.

[0007] One aspect of this disclosure provides an inlet flow equalization system for an electrostatic precipitator, the system comprising a self-adjusting flow equalization plate assembly and a pre-charging device; The self-adjusting flow equalizer assembly includes: The flow equalization plate body is installed at the inlet of the electrostatic precipitator to perform preliminary flow equalization of the flue gas entering the electrostatic precipitator. An adjustment mechanism, disposed on the flow equalization plate body, is used to adjust the angle of the flow equalization plate body; The control unit is used to calculate the target flow equalization plate angle based on the flue gas flow rate and velocity, and to control the adjustment mechanism based on the target flow equalization plate angle. The pre-charge device includes: A pre-charging assembly is installed on the airflow distribution plate for pre-charging dust. The electromagnetic hammer rapping unit is used to perform timed rapping and dust removal on the pre-charged assembly in front of the plate.

[0008] Furthermore, the control unit includes: Sensors are used to detect flue gas flow rate and velocity; The controller is used to receive flue gas flow rate and velocity signals, calculate the target flow equalization plate angle, and output control signals to the mechanism for adjustment.

[0009] Furthermore, the sensor employs a thermal gas mass flow sensor and a Pitot tube flow rate sensor; and / or, The controller has a built-in intelligent control algorithm that calculates the target flow equalization plate angle based on the relative root mean square value of flue gas flow rate, velocity, and uniformity.

[0010] Furthermore, the intelligent control algorithm is expressed as follows:

[0011] In the formula, For the target flow equalizer angle, As the reference angle, For real-time flue gas flow rate, As the baseline flue gas flow rate, This refers to the real-time flue gas velocity. As the reference flue gas velocity, The relative root mean square value is the uniformity value. The relative root mean square value of the target uniformity. , , These are the flow rate influence coefficient, flow velocity influence coefficient, and uniformity influence coefficient, respectively.

[0012] Furthermore, the pre-charge assembly before the board includes: A cathode frame, used to generate corona discharge, forms an anode-cathode pair with the airflow distribution plate; An insulating structure for supporting and insulating the cathode frame.

[0013] Furthermore, the electromagnetic hammer rapping unit is disposed on the top of the pre-charged assembly in front of the plate and the airflow distribution plate.

[0014] Furthermore, the system also includes a power supply unit for supplying power to the pre-charge device and the first electric field of the electrostatic precipitator via a double-knife-switch isolation cabinet.

[0015] Furthermore, the flow equalization plate body is made of high-strength corrosion-resistant alloy material.

[0016] Another aspect of this disclosure provides a method for equalizing the inlet flow of an electrostatic precipitator, based on the system described above, the method comprising: The system detects the flue gas flow rate and velocity and transmits them to the control unit. Based on the flue gas flow rate and velocity, it calculates the target flow equalization plate angle and outputs a control signal to the adjustment mechanism to adjust the angle of the flow equalization plate body. The dust is pre-charged using a pre-charging device so that the charged dust is collected by the airflow distribution plate, and the dust is cleaned by the electromagnetic hammer rapping unit at regular intervals.

[0017] Furthermore, the calculation of the target flow equalization plate angle based on the flue gas flow rate and velocity is expressed by the following formula:

[0018] In the formula, For the target flow equalizer angle, As the reference angle, For real-time flue gas flow rate, As the baseline flue gas flow rate, This refers to the real-time flue gas velocity. As the reference flue gas velocity, The relative root mean square value is the uniformity value. The relative root mean square value of the target uniformity. , , These are the flow rate influence coefficient, flow velocity influence coefficient, and uniformity influence coefficient, respectively.

[0019] This disclosure discloses an inlet flow equalization system and method for an electrostatic precipitator. Based on real-time flue gas flow rate and velocity, it automatically and precisely adjusts the angle of the flow equalization plate to ensure uniform flue gas entry into the electrostatic precipitator. This effectively improves the uneven distribution of inlet flue gas and reduces airflow vortex problems, providing a stable and uniform airflow environment for subsequent dust removal processes. By setting a pre-charging device, pre-charging of dust and partial dust collection are achieved, reducing the dust collection pressure of the subsequent electric field and improving the dust collection efficiency of the preceding electric field. Simultaneously, it reduces the adhesion of substances such as ammonium bisulfate to the electrode plates and wires, significantly improving the overall dust removal efficiency of the electrostatic precipitator. The structure is reasonably designed, easy to operate and maintain, and the dust cleaning operation is highly efficient and reliable. Automated operation can be achieved through the control unit, reducing manual intervention costs. Attached Figure Description

[0020] Figure 1This is a schematic diagram of the structure of an inlet flow equalization system for an electrostatic precipitator according to an embodiment of the present disclosure; Figure 2 This is a schematic flowchart of an electrostatic precipitator inlet flow equalization method according to another embodiment of the present disclosure. Detailed Implementation

[0021] The technical solutions of the embodiments of this disclosure will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only a part of the embodiments of this disclosure, and not all of them. Based on the embodiments of this disclosure, all other embodiments obtained by those of ordinary skill in the art without creative effort are within the scope of protection of this disclosure.

[0022] Furthermore, the described features, structures, or characteristics can be combined in any suitable manner in one or more embodiments. Numerous specific details are provided in the following description to give a thorough understanding of embodiments of this disclosure. However, those skilled in the art will recognize that the technical solutions of this disclosure can be practiced without one or more of the specific details, or other methods, components, apparatuses, steps, etc., can be employed. In other instances, well-known methods, apparatuses, implementations, or operations are not shown or described in detail to avoid obscuring various aspects of this disclosure.

[0023] The flowcharts shown in the accompanying drawings are merely illustrative and do not necessarily include all content and operations / steps, nor do they necessarily have to be performed in the described order. For example, some operations / steps can be broken down, while others can be combined or partially combined; therefore, the actual execution order may change depending on the specific circumstances.

[0024] It should be understood that although the terms first, second, third, etc., may be used in this disclosure to describe various components, these components should not be limited by these terms. These terms are used to distinguish one component from another. Therefore, the first component discussed below may be referred to as the second component without departing from the teachings of this disclosure. As used in this disclosure, the term "and / or" includes all combinations of any one and more of the associated listed items.

[0025] Those skilled in the art will understand that the accompanying drawings are merely schematic diagrams of exemplary embodiments, and the modules or processes in the drawings are not necessarily necessary for implementing this disclosure, and therefore cannot be used to limit the scope of protection of this disclosure.

[0026] like Figure 1 As shown, one embodiment of this disclosure provides an inlet flow equalization system for an electrostatic precipitator, including a self-adjusting flow equalization plate assembly and a pre-charging device.

[0027] The self-adjusting flow equalization plate assembly includes: a flow equalization plate body 1, installed at the inlet 3 of the electrostatic precipitator, for preliminary flow equalization of the flue gas entering the electrostatic precipitator; an adjustment mechanism, disposed on the flow equalization plate body 1, for adjusting the angle of the flow equalization plate body 1; and a control unit, for calculating the target flow equalization plate angle based on the flue gas flow rate and velocity, and controlling the adjustment mechanism based on the target flow equalization plate angle. The pre-charging device includes: a pre-charging component in front of the plate, which is disposed on the airflow distribution plate 4 for pre-charging dust; and an electromagnetic hammer rapping unit 2 for periodically rapping and cleaning the pre-charging component in front of the plate.

[0028] Specifically, the flow equalization plate body 1 is installed behind the bell-shaped opening of the electrostatic precipitator inlet 3. It adopts a louvered structure, composed of multiple adjustable blades, and is customized according to the size and shape of the inlet bell-shaped opening 6 of the electrostatic precipitator to ensure complete coverage of the inlet cross-section. The number of louvered blades is determined according to the inlet size, typically 8-12 blades, each with a width of 150mm-200mm. Each blade is made of high-strength, corrosion-resistant alloy material, providing initial flow equalization for the flue gas entering the electrostatic precipitator. The louvered blades are connected by adjusting rods, enabling multi-blade linkage adjustment, with an adjustment angle range of [range missing]. The temperature range is 30° to +30°, and it can be adaptively adjusted according to the flue gas flow rate and velocity.

[0029] The adjustment mechanism is located on the flow equalization plate body 1 and adopts a high-torque, fast-response electric actuator. It is electrically connected to the control unit and, according to the control signal output by the control unit, precisely adjusts the angle of the flow equalization plate body 1 to the target flow equalization plate within 2 seconds through worm gear transmission or lead screw and nut transmission, with an adjustment accuracy of 0.1°.

[0030] The control unit includes sensors and a controller: the sensors employ high-precision thermal gas mass flow sensors and Pitot tube velocity sensors to detect the inlet flue gas flow rate and velocity in real time, with a sampling frequency of 50 times per second to ensure timely capture of minute changes in flue gas parameters; the controller receives the flue gas flow rate and velocity signals collected by the sensors, and has a built-in microprocessor and intelligent control algorithm to quickly calculate the optimal flow equalizer angle with a calculation response time of less than 0.1 seconds, and finally outputs a control signal based on the calculated optimal flow equalizer angle (target flow equalizer angle).

[0031] The control unit can also integrate a human-machine interface, a data storage module, and a communication module. The human-machine interface allows operators to monitor the real-time operating status of the flow equalization plate body 1, such as flue gas flow rate, velocity, and the current angle of the flow equalization plate body 1, and enables necessary parameter settings and manual intervention. The data storage module stores historical data collected by sensors and the controller's operating parameters, providing data support for equipment maintenance and performance optimization. The communication module can communicate with the electrostatic precipitator's supervisory control and data acquisition (SCADA) system, enabling remote monitoring and centralized management.

[0032] The intelligent control algorithm is based on a multi-parameter coupled model of flue gas flow rate, velocity, and relative root mean square value of uniformity. It achieves accurate calculation of the target flow equalization plate angle through nonlinear fitting, as shown in the following formula:

[0033] In the formula, The target flow equalizer angle (°) has a range of values. 30° to +30°, a positive value indicates that the leaf is tilted upwards, and a negative value indicates that the leaf is tilted downwards; The reference angle (°) is set to 0°, corresponding to the flow equalizer blades being in a horizontal position. Real-time flue gas flow rate (m³) 3 / s), obtained by a thermal gas mass flow sensor; Reference flue gas flow rate (m³) 3 / s), take the rated flow rate under design conditions; The real-time flue gas velocity (m / s) is obtained by a Pitot tube velocity sensor. The reference flue gas velocity (m / s) is taken as the rated velocity value under design conditions; The relative root mean square value (dimensionless) reflects the uniformity of flue gas distribution and is calculated through multi-point flow velocity measurement. The relative root mean square value (dimensionless) for the target uniformity is set to 0.15~0.20 according to industry standards; The flow rate influence coefficient (°) ranges from 5° to 15°, reflecting the degree to which changes in flow rate affect the angle. The velocity influence coefficient (°) ranges from 10° to 20°, reflecting the degree of influence of velocity changes on the angle. The uniformity influence coefficient (°) ranges from 20° to 40°, reflecting the degree of influence of uniformity deviation on the angle.

[0034] Among them, uniformity is relative to the root mean square value Calculated using the following formula:

[0035] In the formula, For the first Flow velocity (m / s) at each measuring point; The total number of measuring points is usually 9 to 16. The average flow velocity is (m / s). .

[0036] The intelligent control algorithm formula achieves accurate fitting of multi-parameter relationships through a combination of nonlinear functions: Flow item Using a logarithmic function can effectively suppress the impact of large fluctuations in flow rate; when the flow rate increases, the angle is adjusted positively; when the flow rate decreases, the angle is adjusted negatively. Flow rate The hyperbolic function is used, which tends to saturate when the flow velocity deviates significantly, thus avoiding over-adjustment. Therefore, the angle is adjusted accordingly when the flow velocity deviates from the reference value. Uniformity term An exponential decay function is used. When the uniformity deviation is large, the angle adjustment range increases; when the uniformity is close to the target value, the adjustment range automatically decreases, thus achieving smooth convergence.

[0037] The pre-charging device includes a pre-charging assembly in front of the plate and an electromagnetic hammer rapping unit 2. The pre-charging assembly in front of the plate is installed on the airflow distribution plate 4 of the electrostatic precipitator and consists of a cathode frame 5 and an insulating structure. Specifically, a conventional multi-layer airflow distribution plate 4 is installed inside the flare 6 of the precipitator, and the cathode frame 5 is installed 800mm before and after the middle layer of the distribution plate. An opening is made in the upper panel of the flare to install a cathode hanging system, and an insulation box and the top electromagnetic hammer rapping unit 2 are installed on top of it. The power supply system is connected to a double-knife switch cabinet and an electric field high-voltage power supply through a high-voltage lead.

[0038] The cathode frame 5 and the airflow distribution plate 4 are combined to form a cathode-cathode pair. When dust-laden flue gas passes through, the dust particles become charged through the cathode frame 5 and are collected by the airflow distribution plate 4. Thus, based on achieving stable flow and secondary distribution of concentration field, it has a certain dust collection capacity. The insulating structure is used to support the cathode frame 5 and insulate it from the airflow distribution plate 4.

[0039] The electromagnetic hammer rapping unit 2 is installed on top of the pre-charged assembly (cathode frame 5 and insulation structure) and the airflow distribution plate 4, respectively, for timed rapping and dust removal. Its rapping strategy can be independently adjusted online, allowing operators to flexibly adjust its lifting height, rapping intensity, rapping frequency, and rapping sequence according to actual conditions. The electromagnetic hammer rapping unit 2 is located outside the flue gas system, unaffected by flue gas interference, making installation and maintenance convenient, with low maintenance costs and a service life of up to 15 years.

[0040] For example, an embodiment of the electrostatic precipitator inlet current equalization system of this disclosure further includes a power supply unit, including a high-voltage power supply and high-voltage leads, for use by the pre-charging device and the first electric field of the electrostatic precipitator. The high-voltage leads are connected to a double-knife-switch disconnector cabinet to ensure that the normal operation of the first electric field is not affected in the event of a malfunction in the pre-charging device, thus enabling the independent operation of the two electrical devices.

[0041] This disclosure discloses an electrostatic precipitator inlet flow equalization system that automatically and precisely adjusts the angle of the flow equalization plate according to the real-time flue gas flow rate and velocity, ensuring that the flue gas enters the electrostatic precipitator evenly. This effectively improves the uneven distribution of inlet flue gas and the problem of airflow vortex, providing a stable and uniform airflow environment for the subsequent dust removal process. By setting up a pre-charging device, the dust is pre-charged and partially collected, reducing the dust collection pressure of the subsequent electric field and improving the dust collection efficiency of the preceding electric field. At the same time, it reduces the adhesion of substances such as ammonium bisulfate to the electrode plates and wires, significantly improving the overall dust removal efficiency of the electrostatic precipitator. Under the same operating conditions, the dust removal efficiency of the electrostatic precipitator is increased by 10% to 20% compared with traditional devices. The system has a reasonable structural design, is easy to operate and maintain, and has efficient and reliable dust removal operation. Automated operation can be achieved through the control unit, reducing the cost of manual intervention. The independent operation of the pre-charging device and the first electric field does not interfere with each other, improving the stability and reliability of operation.

[0042] like Figure 2 As shown, another embodiment of this disclosure provides a method for equalizing the inlet flow of an electrostatic precipitator, based on the electrostatic precipitator inlet flow equalization system described above. The method includes: Step S1: Detect the flue gas flow rate and velocity and transmit them to the control unit. Calculate the target flow equalization plate angle based on the flue gas flow rate and velocity, and output a control signal to the adjustment mechanism to adjust the angle of the flow equalization plate body.

[0043] Specifically, sensors continuously monitor flue gas flow rate and velocity, rapidly transmitting the signals to the controller. Upon receiving the signals, the controller activates its built-in intelligent control algorithm to calculate the optimal flow equalization plate angle (target flow equalization plate angle) within a short time and outputs a control signal. The adjustment mechanism then quickly and precisely adjusts the angle of the flow equalization plate according to the control signal, ensuring that the flue gas enters the operating area of ​​the subsequent pre-charging device evenly.

[0044] Step S2: Use a pre-charging device to pre-charge the dust so that the charged dust is collected by the airflow distribution plate and the electromagnetic hammer rapping unit is used to periodically rappel and clean the dust.

[0045] Specifically, the dust-laden flue gas, after being regulated for flow equalization, enters the pre-charging device area. Under the action of the pre-charging components in front of the plate, the dust is charged and collected by the airflow distribution plate. The top electromagnetic hammer rapping unit cleans the dust periodically according to a pre-set rapping strategy, ensuring the normal operation of the pre-charging components in front of the plate and the airflow distribution plate.

[0046] This disclosure discloses an inlet flow equalization method for an electrostatic precipitator. Based on real-time flue gas flow rate and velocity, the angle of the flow equalization plate is automatically and precisely adjusted to ensure uniform flue gas entry into the electrostatic precipitator. This effectively improves the uneven distribution of inlet flue gas and the problem of airflow vortex, providing a stable and uniform airflow environment for subsequent dust removal processes. By setting up a pre-charging device, pre-charging of dust and partial dust collection are achieved, reducing the dust collection pressure of the subsequent electric field and improving the dust collection efficiency of the preceding electric field. Simultaneously, it reduces the adhesion of substances such as ammonium bisulfate to the electrode plates and wires, significantly improving the overall dust removal efficiency of the electrostatic precipitator. Under the same operating conditions, the dust removal efficiency of the electrostatic precipitator is increased by 10% to 20% compared to traditional devices. The structure is reasonably designed, easy to operate and maintain, and the dust cleaning operation is efficient and reliable. Automated operation can be achieved through the control unit, reducing manual intervention costs. The independent operation of the pre-charging device and the first electric field does not interfere with each other, improving operational stability and reliability.

[0047] It is understood that the above embodiments are merely exemplary embodiments used to illustrate the principles of this disclosure, and this disclosure is not limited thereto. For those skilled in the art, various modifications and improvements can be made without departing from the spirit and substance of this disclosure, and these modifications and improvements are also considered to be within the scope of protection of this disclosure.

Claims

1. A flow equalization system at the inlet of an electrostatic precipitator, characterized in that, The system includes a self-regulating current equalizer assembly and a pre-charging device; The self-adjusting flow equalizer assembly includes: The flow equalization plate body is installed at the inlet of the electrostatic precipitator to perform preliminary flow equalization of the flue gas entering the electrostatic precipitator. An adjustment mechanism, disposed on the flow equalization plate body, is used to adjust the angle of the flow equalization plate body; The control unit is used to calculate the target flow equalization plate angle based on the flue gas flow rate and velocity, and to control the adjustment mechanism based on the target flow equalization plate angle. The pre-charge device includes: A pre-charging assembly is installed on the airflow distribution plate for pre-charging dust. The electromagnetic hammer rapping unit is used to perform timed rapping and dust removal on the pre-charged assembly in front of the plate.

2. The system according to claim 1, characterized in that, The control unit includes: Sensors are used to detect flue gas flow rate and velocity; The controller is used to receive flue gas flow rate and velocity signals, calculate the target flow equalization plate angle, and output control signals to the mechanism for adjustment.

3. The system according to claim 2, characterized in that, The sensors employ a thermal gas mass flow sensor and a Pitot tube flow velocity sensor; and / or, The controller has a built-in intelligent control algorithm that calculates the target flow equalization plate angle based on the relative root mean square value of flue gas flow rate, velocity, and uniformity.

4. The system according to claim 3, characterized in that, The intelligent control algorithm is expressed as follows: In the formula, For the target flow equalizer angle, As the reference angle, For real-time flue gas flow rate, As the baseline flue gas flow rate, This refers to the real-time flue gas velocity. As the reference flue gas velocity, The relative root mean square value is the uniformity value. The relative root mean square value of the target uniformity. , , These are the flow rate influence coefficient, flow velocity influence coefficient, and uniformity influence coefficient, respectively.

5. The system according to claim 1, characterized in that, The pre-charged assembly before the board includes: A cathode frame, used to generate corona discharge, forms an anode-cathode pair with the airflow distribution plate; An insulating structure for supporting and insulating the cathode frame.

6. The system according to claim 1, characterized in that, The electromagnetic hammer vibration unit is located on the top of the pre-charged assembly in front of the plate and the airflow distribution plate.

7. The system according to claim 1, characterized in that, The system also includes a power supply unit for supplying power to the pre-charge device and the first electric field of the electrostatic precipitator via a double-knife-switch isolation cabinet.

8. The system according to any one of claims 1 to 7, characterized in that, The flow equalization plate body is made of high-strength corrosion-resistant alloy material.

9. A method for equalizing the flow at the inlet of an electrostatic precipitator, based on the system described in any one of claims 1 to 8, characterized in that, The method includes: The system detects the flue gas flow rate and velocity and transmits them to the control unit. Based on the flue gas flow rate and velocity, it calculates the target flow equalization plate angle and outputs a control signal to the adjustment mechanism to adjust the angle of the flow equalization plate body. The dust is pre-charged using a pre-charging device so that the charged dust is collected by the airflow distribution plate, and the dust is cleaned by the electromagnetic hammer rapping unit at regular intervals.

10. The method according to claim 9, characterized in that, The angle of the target flow equalizer plate, calculated based on flue gas flow rate and velocity, is expressed by the following formula: In the formula, For the target flow equalizer angle, As the reference angle, For real-time flue gas flow rate, As the baseline flue gas flow rate, This refers to the real-time flue gas velocity. As the reference flue gas velocity, The relative root mean square value is the uniformity value. The relative root mean square value of the target uniformity. , , These are the flow rate influence coefficient, flow velocity influence coefficient, and uniformity influence coefficient, respectively.

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