Pressure determination method for pressure reduction hydrogen transmission of natural gas pipeline

Abstract

The present application relates to pipeline hydrogen transportation technology, and aims to provide a pressure determination method for hydrogen transportation by pressure reduction of natural gas pipeline. The method comprises the following steps: analyzing the crack and defect distribution in the pipeline based on the internal detection result of the pipeline, and determining the risk coefficient of all cracks or defects; selecting three regions with the largest risk coefficient, and cutting off the corresponding pipe sections; conducting blasting test on the cut-off pipe sections by using hydrogen; sampling at the position axially and centrally symmetric to the blasting hole on the cross section of the pipe section, and testing the fracture toughness and fatigue crack propagation rate in the original environment; based on the test result, safety evaluation is conducted on the fatigue life and fracture of the pipeline material, and if the conclusions of the two evaluations both meet the provisions of the mandatory standard, the hydrogen transportation pressure is taken as the maximum operating pressure of the natural gas pipeline in the process of hydrogen transportation by pressure reduction. The present application can quickly locate the most dangerous part of the pipeline after hydrogen mixing, and accurately identify the ultimate bearing capacity of the pipeline while replacing the dangerous pipe section.

Description

Technical Field

[0001] This invention relates to pipeline hydrogen transportation technology, and in particular to a method for determining the pressure of hydrogen transportation via natural gas pipeline with reduced pressure. Background Technology

[0002] Pipeline transportation of hydrogen is the most economical way to achieve large-scale, cross-regional, and long-term hydrogen energy transportation. However, hydrogen pipelines are time-consuming and costly to construct. Utilizing existing natural gas pipelines with hydrogen blending can achieve pipeline transportation of hydrogen energy in a short time, significantly reducing pipeline construction costs and avoiding the abandonment of existing natural gas pipelines during subsequent decarbonization processes.

[0003] When natural gas is blended with hydrogen, hydrogen molecules dissociate into hydrogen atoms on the material surface and penetrate into the metal. Driven by stress, these atoms accumulate towards the crack tip, making the material more prone to fracture. Therefore, for defective natural gas pipelines, the introduction of hydrogen increases the risk of pipeline rupture, seriously endangering equipment operation and the safety of personnel and property. It is therefore necessary to determine whether the pipeline can operate at its original pressure after hydrogen blending, or whether the operating pressure can be reduced to ensure safe gas delivery without replacing the pipeline.

[0004] Although there is currently a considerable amount of data on the mechanical properties of metallic materials in hydrogen environments, it is still difficult to accurately describe the failure behavior of defective pipelines in hydrogen environments based solely on the effects of hydrogen on materials. Furthermore, current standards, such as GB 32167-2015 "Code for Integrity Management of Oil and Gas Transmission Pipelines," only mention that pipelines with unacceptable defects can be operated at reduced pressures, but do not provide a method for calculating the upper limit of operating pressure for existing natural gas pipelines in hydrogen environments. Summary of the Invention

[0005] The technical problem to be solved by the present invention is to overcome the shortcomings of the prior art and provide a method for determining the pressure of hydrogen transportation through a natural gas pipeline with reduced pressure.

[0006] To solve the technical problem, the solution of the present invention is:

[0007] A method for determining the pressure of hydrogen transport via a natural gas pipeline with reduced pressure is provided, comprising the following steps:

[0008] (1) Conduct a comprehensive internal inspection of the natural gas pipeline intended for hydrogen transportation in accordance with the requirements of mandatory standards; analyze the distribution of cracks and defects inside the pipeline based on the inspection results, and determine the hazard factor F of all cracks or defects. H ;

[0009] (2) Selecting the risk factor F in natural gas pipelines HThe three largest areas were used to cut off corresponding pipe sections; hydrogen gas was used as the medium to conduct burst tests on the cut pipe sections to obtain the burst pressure values; assuming the actual operating pressure of the natural gas pipeline is P1, the average burst pressure obtained from the tests is P. b Then the maximum hydrogen transport operating pressure of the natural gas pipeline is: P2 = min{P1, 0.85P} b};

[0010] (3) Take samples at locations axially and centrally symmetrical to the rupture opening on the cross-section of the pipe section, and test the fracture toughness K in situ. IH And testing of fatigue crack propagation rate;

[0011] (4) Based on the test results, a safety evaluation of the fatigue life and fracture of the pipeline material is conducted. If the conclusions of both evaluations meet the requirements of the mandatory standards, then the hydrogen transmission pressure P2 is taken as the maximum operating pressure of the natural gas pipeline during the depressurization hydrogen transmission process.

[0012] As a preferred embodiment of the present invention, in step (1), the hazard factor F of the crack or defect is determined by the following method. H :

[0013] Based on the actual operating pressure P1 of the natural gas pipeline and the material performance parameters, the location A of the detected crack on the failure assessment diagram is calculated. The intersection point of the line connecting the origin O and point A with the failure assessment curve is B; therefore, the hazard factor is F. H =OA / OB, where OA and OB are the lengths of the lines connecting the origin O to points A and B, respectively.

[0014] As a preferred embodiment of the present invention, in step (2), the cut pipe segment contains at least one weld area; the length of the pipe segment is at least 10 times the outer diameter of the pipe segment, and the target defect is located in the middle part of the cut pipe segment.

[0015] As a preferred embodiment of the present invention, in step (2), the cut pipe section is subjected to a burst test in the following manner:

[0016] (a) First, place a high-density polyethylene cylinder inside the pipe section and seal both ends of the pipe section by welding a blind flange. The blind flange is equipped with an air inlet. The diameter of the polyethylene cylinder is 2 to 5 mm smaller than the inner diameter of the pipe section, and its length is at least 10 to 20 mm shorter than the pipe section.

[0017] (b) Hydrogen gas is introduced into the pipeline section through the gas filling port and pressurized to 85% of the operating pressure P1 of the natural gas pipeline. The pressure is maintained for more than 24 hours. The pressure loss during the pressure maintenance period should be less than 5% of the actual operating pressure P1 of the natural gas pipeline.

[0018] (c) Increase the hydrogen pressure in the pipe section at a pressurization rate of less than 1 MPa / h until the pipe section bursts; check whether the burst point is located at the target defect location; if so, the burst pressure test is considered valid; if not, another pipe section needs to be cut for testing; obtain data for at least 3 valid burst pressures, and record the average value as P. b .

[0019] As a preferred embodiment of the present invention, in step (3), conducting the test in an in-situ environment means:

[0020] (1) If pure hydrogen will be transported via natural gas pipeline in the future, hydrogen will be introduced during the test, and the pressure will not be lower than the actual operating pressure P1 of the natural gas pipeline.

[0021] (2) If the hydrogen-blended gas will be transported via natural gas pipeline in the future, the hydrogen-blended gas will be introduced during the test; and the total pressure of the test gas shall not be lower than the actual operating pressure P1 of the natural gas pipeline, and the volume ratio of hydrogen in the test gas shall not be less than the hydrogen blending ratio of the gas to be transported in the future.

[0022] As a preferred embodiment of the present invention, in the fatigue crack propagation rate test of step (3), the stress intensity factor ratio is not greater than the minimum pressure ratio R in the actual operation record of the pipeline. min .

[0023] As a preferred embodiment of the present invention, in step (4), if one of the conclusions of the two evaluations does not comply with the mandatory standard, the hydrogen transmission pressure P2 is multiplied by 0.85, and then the fatigue life evaluation and fracture safety evaluation are carried out again; this operation is repeated until the conclusions of the two evaluations both comply with the regulations, and the hydrogen transmission pressure at this time is taken as the final operating pressure of the natural gas pipeline during the depressurization hydrogen transmission process.

[0024] As a preferred embodiment of the present invention, in step (4), when evaluating fatigue life, the pipeline pressure ratio is not greater than the minimum pressure ratio R in the actual pipeline operation record. min .

[0025] Compared with the prior art, the beneficial effects of the present invention are:

[0026] (1) This invention combines pipeline detection technology and the definition of risk factor to quickly locate the most dangerous part of the pipeline after hydrogen addition.

[0027] (2) This invention conducts hydrogen explosion tests on the most dangerous pipe sections, which can determine the ultimate bearing capacity of the pipeline while replacing the dangerous pipe sections.

[0028] (3) The present invention uses hydrogen as the medium for pipeline burst test, which can more accurately obtain the ultimate bearing capacity of the pipeline. Attached Figure Description

[0029] Figure 1 This is a schematic diagram illustrating the method for calculating the risk factor.

[0030] Figure 2 This is a failure assessment diagram for a crack in an application example.

[0031] Figure 3 This is a failure assessment diagram after updating the crack in the application example.

[0032] Figure 4 This is a graph showing the relationship between crack depth and number of cycles in an application example. Detailed Implementation

[0033] The technical solution of the present invention will now be described in detail with reference to the accompanying drawings.

[0034] The method for determining the pressure of hydrogen transportation via natural gas pipeline depressurization according to the present invention includes the following steps:

[0035] 1. First, in accordance with the requirements of mandatory standards such as GB / T 27699, a comprehensive internal inspection was conducted on the natural gas pipeline intended for hydrogen transportation, recording the distribution and size information of cracks and defects inside the pipeline. Then, referring to relevant standards such as GB / T 19624, API 579, and BS 7910, the distribution of cracks and defects inside the pipeline was calculated based on the inspection results, and the hazard factor F of all cracks or defects was determined. H The risk factor F of cracks or defects H The larger the diameter, the weaker the pipeline area will be under hydrogen transport conditions.

[0036] Specifically, the risk factor F of cracks or defects H This can be determined using the following methods:

[0037] Based on the actual operating pressure P1 of the natural gas pipeline and the material performance parameters, the location A of the detected crack on the failure assessment diagram is calculated. The intersection point of the line connecting the origin O and point A with the failure assessment curve is B; therefore, the hazard factor is F. H =OA / OB, where OA and OB are the lengths of the lines connecting the origin O to points A and B, respectively.

[0038] If the stress-strain curve of the material in a hydrogen environment is unknown, the failure assessment curve can be plotted by referring to the conventional assessment of planar defects in GB / T 19624, the API 579-1 level 2 assessment, and the methods specified in Option 1 of BS 7910. If the stress-strain curve of the material in a hydrogen environment is known, the failure assessment curve can be plotted using the methods specified in API 579-1 level 3 and Option 2 of BS 7910.

[0039] 2. Select the hazard factor F in the natural gas pipeline H The three largest areas are used to cut off the corresponding pipe sections; each cut section should contain at least one weld area. The length of the pipe section should be at least 10 times the outer diameter of the pipe section, and the target defect should be located in the middle of the cut section.

[0040] Using hydrogen as the medium, burst tests were conducted on the selected pipe sections in the following manner to obtain the burst pressure values:

[0041] (a) To reduce gas consumption and the risk of pipeline rupture, a high-density polyethylene cylinder is first placed inside the pipe section. The diameter of the polyethylene cylinder is 2-5 mm smaller than the inner diameter of the pipe section, and its length is at least 10-20 mm shorter than the pipe section. For the intended use, this invention does not have special requirements for the density or other performance parameters of the cylinder material. Then, both ends of the pipe section are sealed by welding blind flanges, with inflation ports provided on the blind flanges.

[0042] (b) Hydrogen gas is introduced into the pipeline section through the gas inlet and pressurized to 85% of the operating pressure P1 of the natural gas pipeline. The pressure is maintained for more than 24 hours. The pressure loss during the pressure maintenance period should be less than 5% of the actual operating pressure P1 of the natural gas pipeline.

[0043] (c) Increase the hydrogen pressure in the pipe section at a rate lower than 1 MPa / h until the pipe section bursts. Check if the burst point is located at the target defect. If so, the burst pressure test is considered valid; if not, another pipe section needs to be cut and tested again. Obtain data for at least 3 valid burst pressures, and record the average value as P. b .

[0044] Assuming the actual operating pressure of the natural gas pipeline is P1, then the maximum hydrogen transport operating pressure of the natural gas pipeline is:

[0045] P2 = min{P1, 0.85P} b}

[0046] 3. Take samples at locations axially symmetrical to the rupture opening on the cross-section of the pipe section, and test the fracture toughness K in situ. IH The fatigue crack propagation rate was tested, as detailed below:

[0047] If pure hydrogen will be transported via natural gas pipeline in the future, hydrogen will be introduced during the test, and the pressure will not be lower than the actual operating pressure P1 of the natural gas pipeline. If hydrogen-blended mixed gas will be transported via natural gas pipeline in the future, hydrogen-blended mixed gas will be introduced during the test. The total pressure of the test gas will not be lower than the actual operating pressure P1 of the natural gas pipeline, and the volume percentage of hydrogen in the test gas will not be less than the hydrogen blending ratio of the mixed gas to be transported in the future.

[0048] In fatigue crack propagation rate testing, the stress intensity factor ratio should not exceed the minimum pressure ratio R recorded in the actual pipeline operation data. min .

[0049] 4. Based on test results, conduct a safety evaluation of the fatigue life and fracture of pipeline materials according to relevant standards such as GB / T 19624, API 579, and BS7910. When evaluating fatigue life, the pipeline pressure ratio should not exceed the minimum pressure ratio R recorded in the actual pipeline operation. min .

[0050] If both evaluations meet the mandatory standards, the hydrogen delivery pressure P2 will be used as the maximum operating pressure of the natural gas pipeline during the depressurization hydrogen delivery process. If either evaluation fails to meet the mandatory standards, the hydrogen delivery pressure P2 will be multiplied by 0.85, and then the fatigue life evaluation and fracture safety evaluation will be conducted again. This process will be repeated until both evaluations meet the requirements, and the hydrogen delivery pressure at this point will be used as the final operating pressure of the natural gas pipeline during the depressurization hydrogen delivery process.

[0051] 5. Before putting natural gas pipelines into actual use, replace the sections of pipeline that have been cut off and the parts of the pipeline whose comprehensive internal inspection results exceed the mandatory standard requirements, so as to avoid putting pipelines with potential hazards into hydrogen transportation operations.

[0052] A specific application example:

[0053] The following example, calculating the delivery pressure of a certain X80 natural gas pipeline after hydrogen addition, further illustrates the invention:

[0054] Taking an X80 natural gas pipeline as an example, the design pressure is 12 MPa. The pipeline has an outer diameter of 1219 mm, a wall thickness of 18.4 mm, and a hydrogen content of 20%. The mechanical properties of the X80 material under these conditions are: yield strength σ... m =660MPa; fracture toughness K 1H =85MPa·m 1 / 2 After hydrogen is added to the pipeline, it still needs to meet 60 years of safe operation requirements. The maximum pressure fluctuation during pipeline operation is expected to be R=P. max / P min =0.8, there will be two pressure fluctuations per day.

[0055] The following is the calculation process for the operating pressure of this natural gas pipeline after hydrogen blending:

[0056] Step 1: Calculate the hazard factor F for each crack. H

[0057] According to the relevant requirements in standard GB / T 27699, a comprehensive internal inspection was carried out on the pipeline. Some of the inspection results are listed below: Crack 1 (depth a1 = 3 mm, length c1 = 9 mm), Crack 2 (depth a2 = 1.5 mm, length c2 = 5 mm).

[0058] The hazard factor F for crack 1 was calculated according to the method in API 579-1 level 2. H as follows:

[0059] First, the stress intensity factor at the crack tip is calculated using the standard formula (KCSCLE1) for calculating the stress intensity factor of a semi-elliptical axial inner surface crack in an internally pressurized cylinder:

[0060]

[0061]

[0062] G0, G1, G2, G3, and G4 can be derived by interpolation using the relevant formulas and tables in the standard.

[0063] The following variables need to be known in this process:

[0064] p is the operating pressure of the pipeline, 12MPa; Ro is the outer diameter of the pipeline, 609.5mm; Ri is the inner diameter of the pipeline, 591.1mm; a is the depth of the crack, 3mm; c is half the crack length, 4.5mm.

[0065] The calculated stress intensity factor at the deepest point of crack 1 is 31.5 MPa·m. 1 / 2

[0066] Next, the reference stress at the crack tip is calculated using the standard formula (RCSCLE1) for calculating the reference stress of a semi-elliptical axial inner surface crack in an internally pressurized cylinder:

[0067]

[0068] In the formula, P is the component of the membrane stress in the pipe. m =pR i / t, the component of bending stress P in the pipe b =p / 2, the calculation methods for other coefficients are as follows:

[0069]

[0070] In the formula, t is the wall thickness of the pipe, which is 18.4 mm.

[0071]

[0072] Shell surface correction factor Ms The calculation uses the net cross-section collapse method:

[0073]

[0074] For M, an axial through-wall crack in a cylinder t The following formula is recommended for calculation:

[0075]

[0076]

[0077] The calculated reference stress at the deepest point of crack 1 is 462 MPa.

[0078] like Figure 2 As shown, for crack 1, its coordinates on the failure assessment diagram are (σ ref / σ m ,K1 / K 1H That is, point A is (0.7, 0.37).

[0079] According to the expression for the plane defect failure assessment curve in GB 19624: y = (1 - 0.14x) 2 )[0.3+0.7exp(-0.65x 6 If the coordinates of the intersection point B of OA and the curve are (1.02, 0.54), then the hazard factor F of crack 1 can be calculated. H =OA / OB=0.686. The hazard coefficient of all cracks detected during pipeline inspection is calculated using this method.

[0080] Step 2: Preliminary determination of the operating pressure of the pipeline after hydrogen addition through explosion experiments.

[0081] Assuming crack 1 corresponds to the highest hazard factor in the pipeline, the calculation of the pipeline operating pressure is demonstrated using crack 1 as an example. A 15m section of the pipeline is cut off from the location of crack 1 for a burst test.

[0082] The pipe was filled with high-density polyethylene cylinders with a diameter of 1200 mm. Then, both ends of the pipe section were sealed by welding blind flanges, with air inlets on the blind flanges. Hydrogen gas was filled into the pipe at a rate of 1 MPa / h to 10.2 MPa, and then the pressure was maintained for 24 hours, ensuring that the pressure loss inside the pipe did not exceed 0.6 MPa during the period.

[0083] Then, hydrogen gas is injected into the pipeline at a rate of 0.5 MPa / h until the pipeline bursts. The burst pressure P of the pipeline is... b The pressure is 13.5 MPa. Therefore, the maximum operating pressure for hydrogen transport in the natural gas pipeline is P2 = min{P1, 0.85P}. b= 11.475 MPa.

[0084] Step 3: Sample and test the mechanical properties of the material under hydrogen oxidative stress.

[0085] Samples were taken at locations axially and centrally symmetrical about the rupture opening on the cross-section of the pipe section, and fracture toughness K was tested in situ. IH The fatigue crack propagation rate was tested in an environment of 9.6 MPa nitrogen + 2.4 MPa hydrogen. The experimental results are as follows:

[0086] Fracture toughness K in hydrogen environment 1H =85MPa·m 1 / 2 ;

[0087] Fatigue crack propagation curve da / dN=3.51×10 -7 ΔK 2.574 ;

[0088] Step 4: Further determine the operating pressure of the test pipeline after hydrogen addition through fatigue assessment.

[0089] According to the fatigue assessment method for pressure vessels in ASME BPVC VIII-3 KD-4, the operating pressure of the pipeline is P = 11.475 MPa. The propagation rates in the depth and length directions of the crack are calculated by the following formula, where dN = 10:

[0090] da / dN=3.51×10 -7 ΔK 2.574

[0091] After each update of the crack size, the pipeline is reassessed based on the failure analysis diagram until the assessment point intersects the failure analysis curve or the crack depth reaches 80% of the pipeline wall thickness. For example... Figure 3 As shown, the critical crack depth 'a' of the pipeline under this operating condition is... c = 7.79mm.

[0092] Calculations show that under this operating pressure, if Figure 4 As shown, min{N a N b 0.5N c The fatigue life of the pipeline is set at 28,000 cycles, which translates to 38.35 years, failing to meet the required fatigue life. Therefore, the operating pressure is further reduced to 85% of the original pressure, i.e., 9.75 MPa. The fatigue life is then recalculated to 55,000 cycles, or 75 years, meeting the fatigue life requirement.

[0093] Therefore, the final calculated operating pressure of the pipeline after hydrogen addition is 9.75 MPa according to this method.

[0094] Step 5: Replace the cut pipe section

[0095] Before putting natural gas pipelines into actual use, the sections that have been cut off and the parts of the pipeline whose comprehensive internal inspection results exceed the mandatory standards should be replaced to avoid putting pipelines with potential hazards into hydrogen transportation operations.

Claims

1. A method for determining the pressure of hydrogen transport via a natural gas pipeline with reduced pressure, characterized in that, Includes the following steps: (1) Conduct a comprehensive internal inspection of the natural gas pipeline intended for hydrogen transportation in accordance with the requirements of mandatory standards; analyze the distribution of defects inside the pipeline based on the inspection results, and determine the risk factor F of all defects. H The defects include cracks, and the mandatory standards include GB / T 19624 and GB / T 27699. (2) Select the hazard factor F in the natural gas pipeline H The three largest areas will have their corresponding pipe sections cut off; Hydrogen gas was used as the medium to conduct burst tests on selected pipe sections, and the burst pressure values ​​were obtained. Assuming the actual operating pressure of the natural gas pipeline is P1, the average burst pressure obtained from the tests is P. b Then the maximum hydrogen transport operating pressure of the natural gas pipeline is: P2 = min{P1, 0.85P} b }; (3) Take samples at locations axially symmetrical to the rupture opening on the cross-section of the pipe section, and test the fracture toughness K in situ. IH And testing of fatigue crack propagation rate; (4) Based on the test results, fatigue life evaluation and fracture safety evaluation of pipeline materials are carried out. If the conclusions of both evaluations meet the requirements of the mandatory standard, the hydrogen transmission pressure P2 shall be taken as the maximum operating pressure of the natural gas pipeline during the depressurization hydrogen transmission process.

2. The method according to claim 1, characterized in that, In step (1), the hazard factor F of the defect is determined by the following method. H : Based on the actual operating pressure P1 of the natural gas pipeline and the material performance parameters, the location A of the detected crack on the failure assessment diagram is calculated. The intersection point of the line connecting the origin O and point A with the failure assessment curve is B; therefore, the hazard factor is F. H =OA / OB, where OA and OB are the lengths of the lines connecting the origin O to points A and B, respectively.

3. The method according to claim 1, characterized in that, In step (2), the cut pipe segment contains at least one weld area; the length of the pipe segment is at least 10 times the outer diameter of the pipe segment, and the target defect is located in the middle of the cut pipe segment.

4. The method according to claim 1, characterized in that, In step (2), the cut pipe section is subjected to a burst test in the following manner: (a) First, place a high-density polyethylene cylinder inside the pipe section and seal both ends of the pipe section by welding a blind flange. The blind flange is equipped with an air inlet. The diameter of the polyethylene cylinder is 2-5 mm smaller than the inner diameter of the pipe section, and the length is 10-20 mm shorter than the pipe section. (b) Hydrogen gas is introduced into the pipeline section through the filling port and pressurized to 85% of the natural gas pipeline operating pressure P1. The pressure is then maintained for at least 24 hours. The pressure loss during the pressure maintenance period should be less than 5% of the actual operating pressure P1 of the natural gas pipeline. (c) Increase the hydrogen pressure in the pipe section at a pressurization rate of less than 1 MPa / h until the pipe section bursts; check whether the burst point is located at the location of the target defect; if so, the burst pressure test is considered valid; if not, another pipe section needs to be cut for testing; obtain data of at least 3 valid burst pressures, and record the average value as P. b .

5. The method according to claim 1, characterized in that, In step (3), conducting the test in situ means: (1) If pure hydrogen will be transported via natural gas pipeline in the future, hydrogen will be introduced during the test, and the pressure will not be lower than the actual operating pressure P1 of the natural gas pipeline. (2) If the hydrogen-blended gas will be transported via natural gas pipeline in the future, the hydrogen-blended gas will be introduced during the test; and the total pressure of the test gas shall not be lower than the actual operating pressure P1 of the natural gas pipeline, and the volume ratio of hydrogen in the test gas shall not be less than the hydrogen blending ratio of the gas to be transported in the future.

6. The method according to claim 1, characterized in that, In the fatigue crack propagation rate test in step (3), the stress intensity factor ratio shall not be greater than the minimum pressure ratio R in the actual pipeline operation record. min .

7. The method according to claim 1, characterized in that, In step (4), if one of the conclusions of the two evaluations does not comply with the mandatory standard, the hydrogen transmission pressure P2 is multiplied by 0.85, and then the fatigue life evaluation and fracture safety evaluation are carried out again. This operation is repeated until the conclusions of the two evaluations both comply with the regulations, and the hydrogen transmission pressure at this time is taken as the final operating pressure of the natural gas pipeline during the depressurization hydrogen transmission process.

8. The method according to claim 1, characterized in that, In step (4), when evaluating fatigue life, the pipeline pressure ratio should not be greater than the minimum pressure ratio R recorded in the actual pipeline operation record. min .

Images