ﺣﻞ ﻣﻌﺎدﻻت ﺿﺮﺑﻪ ﻗﻮچ ﺑﺎ روش وﯾﺴﮑﻮزﯾﺘﻪ ﻓﻮق ﻃﯿﻔﯽ ﭼﺒ ﯿﺸﻒ و ﺿﺮﯾﺐ زﺑﺮي ﻏﯿﺮﻣﺎﻧﺪﮔﺎر

Solving Water Hammer equations by Chebyshev super-spectral viscosity method and unsteady roughness coefficient

ﭘﯿﺶﺑﯿﻨﯽ ﻓﺸﺎر در اﺛﺮ ﭘﺪﯾﺪه ﺿﺮﺑﻪ ﻗﻮچ در ﻃﺮاﺣﯽ ﺧﻂ اﻧﺘﻘﺎل و ﺗﺠﻬﯿﺰات ﻣﻘﺎﺑﻠﻪ ﺑﺎ ﺿﺮﺑﻪ ﻗﻮچ از اﻫﻤﯿﺖ زﯾﺎدي ﺑﺮﺧﻮردار اﺳﺖ. روشﻫﺎي ﻋﺪدي ﻣﺨﺘﻠﻔﯽ ﺑﺮاي ﺣﻞ ﻣﻌﺎدﻻت ﺿﺮﺑﻪ ﻗﻮچ ﺑﻪ ﮐﺎر ﻣﯽرود. در اﯾﻦ ﭘﮋوﻫﺶ روش ﻃﯿﻔﯽ ﭼﺒﯿﺸﻒ و روش وﯾﺴﮑﻮزﯾﺘﻪ ﻓﻮق ﻃﯿﻔﯽ ﭼﺒﯿﺸﻒ ﺑﺮاي ﺣﻞ اﯾﻦ ﻣﻌﺎدﻻت در ﺣﺎﻟﺖ ﺑﺴﺘﻪ ﺷﺪن ﻧﺎﮔﻬﺎﻧﯽ و آﻫﺴﺘﻪ ﺷﯿﺮ ﺑﺎ ﺿﺮﯾﺐ زﺑﺮي ﻣﺎﻧﺪﮔﺎر، ﺷﺒﻪﻣﺎﻧﺪﮔﺎر و ﻏﯿﺮﻣﺎﻧﺪﮔﺎر ﺑﻪ ﮐﺎر رﻓﺘﻪ اﺳﺖ و ﺑﺎ دادهﻫﺎي آزﻣﺎﯾﺸﮕﺎﻫﯽ ﻣﻘﺎﯾﺴﻪ ﺷﺪ. در ﺗﻤﺎم ﺣﺎﻻت ﻧﺘﺎﯾﺞ روش ﻃﯿﻔﯽ ﭼﺒﯿﺸﻒ ﺑﺎ ﺿﺮﯾﺐ زﺑﺮي ﻏﯿﺮﻣﺎﻧﺪﮔﺎر ﻧﺴﺒﺖ ﺑﻪ ﺷﺒﻪ ﻣﺎﻧﺪﮔﺎر و ﻣﺎﻧﺪﮔﺎر ﺑﻪ دادهﻫﺎي آزﻣﺎﯾﺸﮕﺎﻫﯽ ﺑﺴﯿﺎر ﻧﺰدﯾﮑﺘﺮ اﺳﺖ. در ﺣﺎﻟﺖ ﺑﺴﺘﻪ ﺷﺪن ﺳﺮﯾﻊ ﺷﯿﺮ ﻫﺮ دو روش ﻃﯿﻔﯽ ﺑﺎ ﺿﺮﯾﺐ زﺑﺮي ﻏﯿﺮﻣﺎﻧﺪﮔﺎر داراي ﺧﻄﺎي 4درﺻﺪ در ﭘﯿﺶﺑﯿﻨﯽ ﻓﺸﺎر و ﻧﺘﺎﯾﺞ ﻫﺮ دو روش ﺑﺮﻫﻢ ﻣﻨﻄﺒﻖ ﻫﺴﺘﻨﺪ. روشﻫﺎي ﻃﯿﻔﯽ ﻓﺸﺎر ﮐﻤﺘﺮ از دادهﻫﺎي آزﻣﺎﯾﺸﮕﺎﻫﯽ را ﻣﺤﺎﺳﺒﻪ ﻣﯽﮐﻨﺪ. در اﯾﻦ ﺣﺎﻟﺖ ﻧﺘﺎﯾﺞ روش ﺗﻔﺎﺿﻞ ﻣﺤﺪود ﻧﯿﺰ داراي ﺧﻄﺎي ﺣﺪود 4درﺻﺪ و ﻓﺸﺎر را ﺑﯿﺸﺘﺮ از ﻓﺸﺎر واﻗﻌﯽ ﻣﺤﺎﺳﺒﻪ ﻣﯽﮐﻨﺪ. اﻣﺎ در ﺣﺎﻟﺖ ﺑﺴﺘﻪ ﺷﺪن آﻫﺴﺘﻪ ﺷﯿﺮ ﺑﺎ ﺿﺮﯾﺐ زﺑﺮي ﻏﯿﺮﻣﺎﻧﺪﮔﺎر، ﺗﻤﺎم روشﻫﺎ در ﻣﺤﺎﺳﺒﻪ ﻓﺸﺎر داراي ﺣﺪاﮐﺜﺮ ﺧﻄﺎي 7درﺻﺪ در اﻧﺘﻬﺎي ﻟﻮﻟﻪ و 2/1درﺻﺪ در وﺳﻂ ﻟﻮﻟﻪ ﻫﺴﺘﻨﺪ و ﻣﻘﺎدﯾﺮ ﻓﺸﺎر ﺣﺪاﮐﺜﺮ 0/0224 ﺛﺎﻧﯿﻪ دﯾﺮﺗﺮ از زﻣﺎن واﻗﻌﯽ ﭘﯿﺶﺑﯿﻨﯽ ﻣﯽﺷﻮد. روش ﻃﯿﻔﯽ ﭼﺒﯿﺸﻒ ﺑﺎ ﺗﻌﺪاد ﮔﺮه و زﻣﺎن ﻣﺤﺎﺳﺒﺎت ﮐﻤﺘﺮ ﻧﺴﺒﺖ ﺑﻪ روش ﻫﺎي ﻋﺪدي روﺷﯽ ﮐﺎرآﻣﺪ در ﺣﻞ ﻣﻌﺎدﻻت ﺿﺮﺑﻪ ﻗﻮچ اﺳﺖ.

Predicting the impact due to the Water Hammer phenomenon is very important in the design of the pipe line and Water Hammer control equipment. Different numerical methods are used to solve Water Hammer equations. In this research, Chebyshev spectral method and Chebyshev super-spectral viscosity method have been used to solve these equations in the case of sudden and slow closing of the valve with steady, quasisteady and unsteady roughness coefficient and were compared with experimental data. In the case of rapid closing of the valve, both spectral methods with unsteady roughness coefficient have high accuracy in predicting the pressure and the results of both methods are same. However, in the case of slow closing of the valve with unsteady roughness coefficient, at the beginning of the Water Hammer, there is an error in calculating the minimum and maximum pressure, and over time, the error decreases and the pressure values are predicted later than the real time. Comparison of the results of Chebyshev super spectral viscosity method with the finite difference method shows that the spectral method has a higher accuracy in fast closing of the valve than the finite difference method. While in the case of slow closing of the valve at the beginning of the Water Hammer, the finite difference method has a higher accuracy in calculating the minimum and maximum pressure values. But over time, the results of the Chebyshev viscosity spectral method conform to the results of the finite difference method, and both methods predict pressure values later.