The Darcy - Weisbach formula is traditionally used in the calculation of pipelines through which oil and petroleum products with small additives of polymers are pumped. To calculate the coefficient of hydraulic resistance, it offers a large number of computational dependencies. The article provides a critical analysis of them. The formulas that were obtained as a result of experiments on water are not suitable for solving the problems of pipeline transportation of oil and oil products. All the others are not fully theoretical, because they contain empirical coefficients, which either have to be refined for each pair of "liquid-additive" according to experimental data, or calculated from the approximation dependencies obtained for the conditions of the performed experiments. In principle, most of the described formulas can be used equally. However, some of them are not convenient for solving theoretical problems of pipeline transport of oil and oil products, because they are transcendental. Therefore, according to the authors, the most preferred formula, in which the coefficient of hydraulic resistance when pumping oil and petroleum products with small additives of polymers is presented as a product of a similar coefficient when pumping without polymers in the form of L.S. Leibenzon and correction function, taking into account the concentration of the polymer and the degree of its impact on the resistance of the pipeline. This record of the calculation formula allowed us to show that for the hydraulic calculation of pipelines for pumping oil and petroleum products with small additives of polymers, the generalized formula of L.S. Leibenzon can be used. Moreover, the coefficient β in it is equal to the product of a similar coefficient when pumping oil and oil products without polymer additives by a correction function that takes into account the polymer concentration, the degree of turbulence development and other factors. Since the other has not yet been established, the value of another L.S. Leibenson coefficient m and the methods for calculating the transition Reynolds numbers when using small polymer additives can be considered the same as in the case of pumping without them.

References

1. Bulina I.G., Dinaburg L.S., Magomedov A.D., Bakaradzhieva V.I., Possible ways to reduce the hydrodynamic resistance during flow in turbulent pipes (In Russ.), Neftyanoe khozyaystvo = Oil Industry, 1971, no. 6, pp. 27–30.

2. Walsh M., Theory of drag reduction in dilute high polymer flows, Trans. Soc. Rheal., 1978, V. 27, pp. 134–137.

3. Barenblatt G.I., Bulina I.G., Zel'dovich Ya.B. et al., One possible mechanism of the effect of small additions of high-molecular composition on turbulence (In Russ.), PMTF, 1965, no. 5, pp. 147–148.

4. Johnson B., Barchi R., Effect of drag reducing additives on boundary–layer turbulence, Journal of hydromantic, 1968, V. 2, pp. 108–110.

5. Amfilokhiev V.B., Artyushkov L.S., Similarity criteria for turbulent flows of dilute polymer solutions and generalized dependence for the coefficient of friction (In Russ.), Izvestiya RAN. Ser. Mekhanika zhidkosti i gaza, 1998, no. 4, pp. 191–196.

6. Lur'e M.V., Hydraulic calculations for pumping diesel fuels with anti-turbulent additives (In Russ.), Transport i khranenie nefteproduktov, 1996, no. 10–11, pp. 18–20.

7. Lur'e M.V., Golunov N.N., Application of bench test results of small antiturbulent additives for industrial pipeline hydraulic analysis (In Russ.), Nauka i tehnologii truboprovodnogo transporta nefti i nefteproduktov = Science & Technologies: Oil and Oil Products Pipeline Transportation, 2016, no. 4, pp. 32–37.

8. Eroshkina I.I., Povyshenie propusknoy sposobnosti magistral'nykh nefteproduktoprovodov na osnove primeneniya protivoturbulentnykh prisadok (Increasing the throughput of main oil product pipelines based on the use of anti-turbulent additives): thesis of candidate of technical science, Moscow, 2003.

9. Khusseyn M.N., Uluchshenie parametrov raboty nefteprovodov putem primeneniya protivoturbulentnykh prisadok (Improving the parameters of oil pipelines by using anti-turbulent additives): thesis of candidate of technical science, Ufa, 2009.

10. Muratova V.I., Nechval' A.M., The choice of the formula for calculating the coefficient of hydraulic resistance when using anti-turbulent additives (In Russ.), Transport i khranenie nefteproduktov, 2008, no. 2, pp. 11–13.

11. Lowe R., The turbulent shear flow of dilute polymer solution a long chain polymers: A thesis presented for the degree of Master of Engineering at University of Liverpool, 1969, no. 7, 115 p.

12. Sedov L.I., Vasetskaya V.A., Ioselevich V.A., Pilipenko V.N., O snizhenii gidrodinamicheskogo soprotivleniya dobavkami polimerov (On the reduction of hydrodynamic resistance by polymer additives), In: Mekhanika turbulentnykh potokov (Mechanics of turbulent flows), Moscow: Nauka Publ., 1980, pp. 7–29.

13. Gorin Ya., Norberi D., Turbulent flow of dilute polymer solutions (In Russ.), Inzhenerno-fizicheskiy zhurnal, 1995, V. 27, no. 5, pp. 830–838.

14. Corless R.M., Connet G.H., Hare D.E. et al., On the Lambert W function, Advance Computational Maths, 1996, V. 5, pp. 329–359.

15. Gol'yanov A.I., Gol'yanov A.A., Mikhaylov D.A. et al., Trunk oil pipeline work specifics with anti-turbulent additive application (In Russ.), Nauka i tehnologii truboprovodnogo transporta nefti i nefteproduktov = Science & Technologies: Oil and Oil Products Pipeline Transportation, 2013, no. 2, pp. 36–43.

References

1. Bulina I.G., Dinaburg L.S., Magomedov A.D., Bakaradzhieva V.I., Possible ways to reduce the hydrodynamic resistance during flow in turbulent pipes (In Russ.), Neftyanoe khozyaystvo = Oil Industry, 1971, no. 6, pp. 27–30.

2. Walsh M., Theory of drag reduction in dilute high polymer flows, Trans. Soc. Rheal., 1978, V. 27, pp. 134–137.

3. Barenblatt G.I., Bulina I.G., Zel'dovich Ya.B. et al., One possible mechanism of the effect of small additions of high-molecular composition on turbulence (In Russ.), PMTF, 1965, no. 5, pp. 147–148.

4. Johnson B., Barchi R., Effect of drag reducing additives on boundary–layer turbulence, Journal of hydromantic, 1968, V. 2, pp. 108–110.

5. Amfilokhiev V.B., Artyushkov L.S., Similarity criteria for turbulent flows of dilute polymer solutions and generalized dependence for the coefficient of friction (In Russ.), Izvestiya RAN. Ser. Mekhanika zhidkosti i gaza, 1998, no. 4, pp. 191–196.

6. Lur'e M.V., Hydraulic calculations for pumping diesel fuels with anti-turbulent additives (In Russ.), Transport i khranenie nefteproduktov, 1996, no. 10–11, pp. 18–20.

7. Lur'e M.V., Golunov N.N., Application of bench test results of small antiturbulent additives for industrial pipeline hydraulic analysis (In Russ.), Nauka i tehnologii truboprovodnogo transporta nefti i nefteproduktov = Science & Technologies: Oil and Oil Products Pipeline Transportation, 2016, no. 4, pp. 32–37.

8. Eroshkina I.I., Povyshenie propusknoy sposobnosti magistral'nykh nefteproduktoprovodov na osnove primeneniya protivoturbulentnykh prisadok (Increasing the throughput of main oil product pipelines based on the use of anti-turbulent additives): thesis of candidate of technical science, Moscow, 2003.

9. Khusseyn M.N., Uluchshenie parametrov raboty nefteprovodov putem primeneniya protivoturbulentnykh prisadok (Improving the parameters of oil pipelines by using anti-turbulent additives): thesis of candidate of technical science, Ufa, 2009.

10. Muratova V.I., Nechval' A.M., The choice of the formula for calculating the coefficient of hydraulic resistance when using anti-turbulent additives (In Russ.), Transport i khranenie nefteproduktov, 2008, no. 2, pp. 11–13.

11. Lowe R., The turbulent shear flow of dilute polymer solution a long chain polymers: A thesis presented for the degree of Master of Engineering at University of Liverpool, 1969, no. 7, 115 p.

12. Sedov L.I., Vasetskaya V.A., Ioselevich V.A., Pilipenko V.N., O snizhenii gidrodinamicheskogo soprotivleniya dobavkami polimerov (On the reduction of hydrodynamic resistance by polymer additives), In: Mekhanika turbulentnykh potokov (Mechanics of turbulent flows), Moscow: Nauka Publ., 1980, pp. 7–29.

13. Gorin Ya., Norberi D., Turbulent flow of dilute polymer solutions (In Russ.), Inzhenerno-fizicheskiy zhurnal, 1995, V. 27, no. 5, pp. 830–838.

14. Corless R.M., Connet G.H., Hare D.E. et al., On the Lambert W function, Advance Computational Maths, 1996, V. 5, pp. 329–359.

15. Gol'yanov A.I., Gol'yanov A.A., Mikhaylov D.A. et al., Trunk oil pipeline work specifics with anti-turbulent additive application (In Russ.), Nauka i tehnologii truboprovodnogo transporta nefti i nefteproduktov = Science & Technologies: Oil and Oil Products Pipeline Transportation, 2013, no. 2, pp. 36–43.

The Darcy - Weisbach formula is traditionally used in the calculation of pipelines through which oil and petroleum products with small additives of polymers are pumped. To calculate the coefficient of hydraulic resistance, it offers a large number of computational dependencies. The article provides a critical analysis of them. The formulas that were obtained as a result of experiments on water are not suitable for solving the problems of pipeline transportation of oil and oil products. All the others are not fully theoretical, because they contain empirical coefficients, which either have to be refined for each pair of "liquid-additive" according to experimental data, or calculated from the approximation dependencies obtained for the conditions of the performed experiments. In principle, most of the described formulas can be used equally. However, some of them are not convenient for solving theoretical problems of pipeline transport of oil and oil products, because they are transcendental. Therefore, according to the authors, the most preferred formula, in which the coefficient of hydraulic resistance when pumping oil and petroleum products with small additives of polymers is presented as a product of a similar coefficient when pumping without polymers in the form of L.S. Leibenzon and correction function, taking into account the concentration of the polymer and the degree of its impact on the resistance of the pipeline. This record of the calculation formula allowed us to show that for the hydraulic calculation of pipelines for pumping oil and petroleum products with small additives of polymers, the generalized formula of L.S. Leibenzon can be used. Moreover, the coefficient β in it is equal to the product of a similar coefficient when pumping oil and oil products without polymer additives by a correction function that takes into account the polymer concentration, the degree of turbulence development and other factors. Since the other has not yet been established, the value of another L.S. Leibenson coefficient m and the methods for calculating the transition Reynolds numbers when using small polymer additives can be considered the same as in the case of pumping without them.

References

1. Bulina I.G., Dinaburg L.S., Magomedov A.D., Bakaradzhieva V.I., Possible ways to reduce the hydrodynamic resistance during flow in turbulent pipes (In Russ.), Neftyanoe khozyaystvo = Oil Industry, 1971, no. 6, pp. 27–30.

2. Walsh M., Theory of drag reduction in dilute high polymer flows, Trans. Soc. Rheal., 1978, V. 27, pp. 134–137.

3. Barenblatt G.I., Bulina I.G., Zel'dovich Ya.B. et al., One possible mechanism of the effect of small additions of high-molecular composition on turbulence (In Russ.), PMTF, 1965, no. 5, pp. 147–148.

4. Johnson B., Barchi R., Effect of drag reducing additives on boundary–layer turbulence, Journal of hydromantic, 1968, V. 2, pp. 108–110.

5. Amfilokhiev V.B., Artyushkov L.S., Similarity criteria for turbulent flows of dilute polymer solutions and generalized dependence for the coefficient of friction (In Russ.), Izvestiya RAN. Ser. Mekhanika zhidkosti i gaza, 1998, no. 4, pp. 191–196.

6. Lur'e M.V., Hydraulic calculations for pumping diesel fuels with anti-turbulent additives (In Russ.), Transport i khranenie nefteproduktov, 1996, no. 10–11, pp. 18–20.

7. Lur'e M.V., Golunov N.N., Application of bench test results of small antiturbulent additives for industrial pipeline hydraulic analysis (In Russ.), Nauka i tehnologii truboprovodnogo transporta nefti i nefteproduktov = Science & Technologies: Oil and Oil Products Pipeline Transportation, 2016, no. 4, pp. 32–37.

8. Eroshkina I.I., Povyshenie propusknoy sposobnosti magistral'nykh nefteproduktoprovodov na osnove primeneniya protivoturbulentnykh prisadok (Increasing the throughput of main oil product pipelines based on the use of anti-turbulent additives): thesis of candidate of technical science, Moscow, 2003.

9. Khusseyn M.N., Uluchshenie parametrov raboty nefteprovodov putem primeneniya protivoturbulentnykh prisadok (Improving the parameters of oil pipelines by using anti-turbulent additives): thesis of candidate of technical science, Ufa, 2009.

10. Muratova V.I., Nechval' A.M., The choice of the formula for calculating the coefficient of hydraulic resistance when using anti-turbulent additives (In Russ.), Transport i khranenie nefteproduktov, 2008, no. 2, pp. 11–13.

11. Lowe R., The turbulent shear flow of dilute polymer solution a long chain polymers: A thesis presented for the degree of Master of Engineering at University of Liverpool, 1969, no. 7, 115 p.

12. Sedov L.I., Vasetskaya V.A., Ioselevich V.A., Pilipenko V.N., O snizhenii gidrodinamicheskogo soprotivleniya dobavkami polimerov (On the reduction of hydrodynamic resistance by polymer additives), In: Mekhanika turbulentnykh potokov (Mechanics of turbulent flows), Moscow: Nauka Publ., 1980, pp. 7–29.

13. Gorin Ya., Norberi D., Turbulent flow of dilute polymer solutions (In Russ.), Inzhenerno-fizicheskiy zhurnal, 1995, V. 27, no. 5, pp. 830–838.

14. Corless R.M., Connet G.H., Hare D.E. et al., On the Lambert W function, Advance Computational Maths, 1996, V. 5, pp. 329–359.

15. Gol'yanov A.I., Gol'yanov A.A., Mikhaylov D.A. et al., Trunk oil pipeline work specifics with anti-turbulent additive application (In Russ.), Nauka i tehnologii truboprovodnogo transporta nefti i nefteproduktov = Science & Technologies: Oil and Oil Products Pipeline Transportation, 2013, no. 2, pp. 36–43.

References

1. Bulina I.G., Dinaburg L.S., Magomedov A.D., Bakaradzhieva V.I., Possible ways to reduce the hydrodynamic resistance during flow in turbulent pipes (In Russ.), Neftyanoe khozyaystvo = Oil Industry, 1971, no. 6, pp. 27–30.

2. Walsh M., Theory of drag reduction in dilute high polymer flows, Trans. Soc. Rheal., 1978, V. 27, pp. 134–137.

3. Barenblatt G.I., Bulina I.G., Zel'dovich Ya.B. et al., One possible mechanism of the effect of small additions of high-molecular composition on turbulence (In Russ.), PMTF, 1965, no. 5, pp. 147–148.

4. Johnson B., Barchi R., Effect of drag reducing additives on boundary–layer turbulence, Journal of hydromantic, 1968, V. 2, pp. 108–110.

5. Amfilokhiev V.B., Artyushkov L.S., Similarity criteria for turbulent flows of dilute polymer solutions and generalized dependence for the coefficient of friction (In Russ.), Izvestiya RAN. Ser. Mekhanika zhidkosti i gaza, 1998, no. 4, pp. 191–196.

6. Lur'e M.V., Hydraulic calculations for pumping diesel fuels with anti-turbulent additives (In Russ.), Transport i khranenie nefteproduktov, 1996, no. 10–11, pp. 18–20.

7. Lur'e M.V., Golunov N.N., Application of bench test results of small antiturbulent additives for industrial pipeline hydraulic analysis (In Russ.), Nauka i tehnologii truboprovodnogo transporta nefti i nefteproduktov = Science & Technologies: Oil and Oil Products Pipeline Transportation, 2016, no. 4, pp. 32–37.

8. Eroshkina I.I., Povyshenie propusknoy sposobnosti magistral'nykh nefteproduktoprovodov na osnove primeneniya protivoturbulentnykh prisadok (Increasing the throughput of main oil product pipelines based on the use of anti-turbulent additives): thesis of candidate of technical science, Moscow, 2003.

9. Khusseyn M.N., Uluchshenie parametrov raboty nefteprovodov putem primeneniya protivoturbulentnykh prisadok (Improving the parameters of oil pipelines by using anti-turbulent additives): thesis of candidate of technical science, Ufa, 2009.

10. Muratova V.I., Nechval' A.M., The choice of the formula for calculating the coefficient of hydraulic resistance when using anti-turbulent additives (In Russ.), Transport i khranenie nefteproduktov, 2008, no. 2, pp. 11–13.

11. Lowe R., The turbulent shear flow of dilute polymer solution a long chain polymers: A thesis presented for the degree of Master of Engineering at University of Liverpool, 1969, no. 7, 115 p.

12. Sedov L.I., Vasetskaya V.A., Ioselevich V.A., Pilipenko V.N., O snizhenii gidrodinamicheskogo soprotivleniya dobavkami polimerov (On the reduction of hydrodynamic resistance by polymer additives), In: Mekhanika turbulentnykh potokov (Mechanics of turbulent flows), Moscow: Nauka Publ., 1980, pp. 7–29.

13. Gorin Ya., Norberi D., Turbulent flow of dilute polymer solutions (In Russ.), Inzhenerno-fizicheskiy zhurnal, 1995, V. 27, no. 5, pp. 830–838.

14. Corless R.M., Connet G.H., Hare D.E. et al., On the Lambert W function, Advance Computational Maths, 1996, V. 5, pp. 329–359.

15. Gol'yanov A.I., Gol'yanov A.A., Mikhaylov D.A. et al., Trunk oil pipeline work specifics with anti-turbulent additive application (In Russ.), Nauka i tehnologii truboprovodnogo transporta nefti i nefteproduktov = Science & Technologies: Oil and Oil Products Pipeline Transportation, 2013, no. 2, pp. 36–43.