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P✐♣❡ ❋❧♦✇s ■■ P✐♣❡ ✐♥❧❡t ❍❡❛❞ ▲♦ss❡s ❚✉r❜✉❧❡♥t ❤❡❛❞ ❧♦ss ❈♦rr❡❧❛t✐♦♥s P✐♣❡ ❋❧♦✇s ■■ ❢♦r ❢ ❙❖❊✷✶✺✻✿ ❋❧✉✐❞s ▲❡❝t✉r❡ ✸ ▼✐♥♦r ❧♦ss❡s ❚②♣❡s ♦❢ ♣r♦❜❧❡♠

P✐♣❡ ❋❧♦✇s ■■ P✐♣❡ ✐♥❧❡t P✐♣❡ ✐♥❧❡t ❍❡❛❞ ▲♦ss❡s ❚✉r❜✉❧❡♥t ❋❧♦✇ ❛❧♦♥❣ ❛ ♣✐♣❡ ❢r♦♠ ❛♥ ✐♥❧❡t ✿ ❤❡❛❞ ❧♦ss ❈♦rr❡❧❛t✐♦♥s ❢♦r ❢ ▼✐♥♦r ❧♦ss❡s ❚②♣❡s ♦❢ ♣r♦❜❧❡♠ Inlet section Fully developed flow ✎ ■♥❧❡t s❡❝t✐♦♥ ④ ✌♦✇ ♣r♦☞❧❡ ❛❧t❡rs ✭t♦ ❜❡❝♦♠❡ ♣❛r❛❜♦❧✐❝✮ ✎ ❋✉❧❧② ❞❡✈❡❧♦♣❡❞ ✌♦✇ ④ ✌♦✇ ♣r♦☞❧❡ r❡♠❛✐♥s ❝♦♥st❛♥t

P✐♣❡ ❋❧♦✇s ■■ P✐♣❡ ✐♥❧❡t ❙t❛rt ❜② ❝♦♥s✐❞❡r✐♥❣ ❢✉❧❧② ❞❡✈❡❧♦♣❡❞ ✌♦✇✳ ❆♣♣❧② ❇❡r♥♦✉❧❧✐✬s ❍❡❛❞ ▲♦ss❡s t❤❡♦r❡♠ ♦♥ ❛ str❡❛♠❧✐♥❡ ❛❧♦♥❣ t❤❡ ♣✐♣❡ ✿ ❚✉r❜✉❧❡♥t ❤❡❛❞ ❧♦ss ✔ ♣ ✔ ♣ ✚ ❣ ✰ ❱ ✷ ✚ ❣ ✰ ❱ ✷ ✕ ✕ ❈♦rr❡❧❛t✐♦♥s ✷ ❣ ✰ ❤ ❂ ✷ ❣ ✰ ❤ ❢♦r ❢ ▼✐♥♦r ❧♦ss❡s ❆ ❇ ❚②♣❡s ♦❢ ♣r♦❜❧❡♠ ✶ ❍♦r✐③♦♥t❛❧ ♣✐♣❡ ✮ ✁ ❤ ❂ ✵ ✷ ❱ ❆ ❂ ❱ ❇ ✱ s♦ ✁✭ ❱ ✷ ❂ ✷ ❣ ✮ ❂ ✵ ✸ ✿ ✿ ✿ t❤✉s ✁ ♣ ❂ ✵ ✐✳❡✳ ♥♦ ✇♦r❦ ✐s ♥❡❝❡ss❛r② t♦ ♣✉♠♣ ✇❛t❡r ❛❧♦♥❣ ❛ ♣✐♣❡ ✿ ✿ ✿ ❄

P✐♣❡ ❋❧♦✇s ■■ ❍❡❛❞ ▲♦ss❡s P✐♣❡ ✐♥❧❡t ❍❡❛❞ ▲♦ss❡s ❚✉r❜✉❧❡♥t ❤❡❛❞ ❧♦ss ■♥tr♦❞✉❝❡ ❛♥ ❡①tr❛ t❡r♠ ❍ ▲ ✐♥t♦ ❇❡r♥♦✉❧❧✐ t♦ r❡♣r❡s❡♥t t❤❡ ❤❡❛❞ ❈♦rr❡❧❛t✐♦♥s ❢♦r ❢ ❧♦ss ✐♥ t❤❡ ♣✐♣❡✳ ◆♦✇ ✿ ▼✐♥♦r ❧♦ss❡s ✔ ♣ ✔ ♣ ✚ ❣ ✰ ❱ ✷ ✚ ❣ ✰ ❱ ✷ ✕ ✕ ❚②♣❡s ♦❢ ✷ ❣ ✰ ❤ ❂ ✷ ❣ ✰ ❤ ✰ ❍ ▲ ♣r♦❜❧❡♠ ❆ ❇ ❚❤✉s ✁ ♣ ✚ ❣ ❂ ❍ ▲ ✐✳❡✳ ✇❡ ♥❡❡❞ ❛ ♣r❡ss✉r❡ ❞✐☛❡r❡♥❝❡ ✁ ♣ ❢♦r ✇❛t❡r t♦ ✌♦✇✳

✎ ♣✐♣❡ ❧❡♥❣t❤ ▲ ✎ ♣✐♣❡ ❞✐❛♠❡t❡r ❉ ✎ ✌♦✇ r❛t❡ ◗ ❝ ▲◗ ✭✰❡♥tr❛♥❝❡ ❡☛❡❝ts✮ ❉ ✹ ❍♦✇❡✈❡r t❤✐s ❝❤❛♥❣❡s ❢♦r t✉r❜✉❧❡♥t ✌♦✇✳ P✐♣❡ ❋❧♦✇s ■■ P✐♣❡ ✐♥❧❡t ❍❡❛❞ ▲♦ss❡s ❊①♣❡r✐♠❡♥t❛❧❧②✱ ♣r❡ss✉r❡ ❞r♦♣ ❞❡♣❡♥❞s ♦♥ ❚✉r❜✉❧❡♥t ❤❡❛❞ ❧♦ss ❈♦rr❡❧❛t✐♦♥s ❢♦r ❢ ▼✐♥♦r ❧♦ss❡s ❚②♣❡s ♦❢ ♣r♦❜❧❡♠ ✐✳❡✳ ✁ ♣ ❂

❝ ▲◗ ✭✰❡♥tr❛♥❝❡ ❡☛❡❝ts✮ ❉ ✹ ❍♦✇❡✈❡r t❤✐s ❝❤❛♥❣❡s ❢♦r t✉r❜✉❧❡♥t ✌♦✇✳ P✐♣❡ ❋❧♦✇s ■■ P✐♣❡ ✐♥❧❡t ❍❡❛❞ ▲♦ss❡s ❊①♣❡r✐♠❡♥t❛❧❧②✱ ♣r❡ss✉r❡ ❞r♦♣ ❞❡♣❡♥❞s ♦♥ ❚✉r❜✉❧❡♥t ❤❡❛❞ ❧♦ss ✎ ♣✐♣❡ ❧❡♥❣t❤ ▲ ❈♦rr❡❧❛t✐♦♥s ✎ ♣✐♣❡ ❞✐❛♠❡t❡r ❉ ❢♦r ❢ ▼✐♥♦r ❧♦ss❡s ✎ ✌♦✇ r❛t❡ ◗ ❚②♣❡s ♦❢ ♣r♦❜❧❡♠ ✐✳❡✳ ✁ ♣ ❂

❍♦✇❡✈❡r t❤✐s ❝❤❛♥❣❡s ❢♦r t✉r❜✉❧❡♥t ✌♦✇✳ P✐♣❡ ❋❧♦✇s ■■ P✐♣❡ ✐♥❧❡t ❍❡❛❞ ▲♦ss❡s ❊①♣❡r✐♠❡♥t❛❧❧②✱ ♣r❡ss✉r❡ ❞r♦♣ ❞❡♣❡♥❞s ♦♥ ❚✉r❜✉❧❡♥t ❤❡❛❞ ❧♦ss ✎ ♣✐♣❡ ❧❡♥❣t❤ ▲ ❈♦rr❡❧❛t✐♦♥s ✎ ♣✐♣❡ ❞✐❛♠❡t❡r ❉ ❢♦r ❢ ▼✐♥♦r ❧♦ss❡s ✎ ✌♦✇ r❛t❡ ◗ ❚②♣❡s ♦❢ ♣r♦❜❧❡♠ ✐✳❡✳ ✁ ♣ ❂ ❝ ▲◗ ✭✰❡♥tr❛♥❝❡ ❡☛❡❝ts✮ ❉ ✹

P✐♣❡ ❋❧♦✇s ■■ P✐♣❡ ✐♥❧❡t ❍❡❛❞ ▲♦ss❡s ❊①♣❡r✐♠❡♥t❛❧❧②✱ ♣r❡ss✉r❡ ❞r♦♣ ❞❡♣❡♥❞s ♦♥ ❚✉r❜✉❧❡♥t ❤❡❛❞ ❧♦ss ✎ ♣✐♣❡ ❧❡♥❣t❤ ▲ ❈♦rr❡❧❛t✐♦♥s ✎ ♣✐♣❡ ❞✐❛♠❡t❡r ❉ ❢♦r ❢ ▼✐♥♦r ❧♦ss❡s ✎ ✌♦✇ r❛t❡ ◗ ❚②♣❡s ♦❢ ♣r♦❜❧❡♠ ✐✳❡✳ ✁ ♣ ❂ ❝ ▲◗ ✭✰❡♥tr❛♥❝❡ ❡☛❡❝ts✮ ❉ ✹ ❍♦✇❡✈❡r t❤✐s ❝❤❛♥❣❡s ❢♦r t✉r❜✉❧❡♥t ✌♦✇✳

P✐♣❡ ❋❧♦✇s ■■ ❚✉r❜✉❧❡♥t ❤❡❛❞ ❧♦ss P✐♣❡ ✐♥❧❡t ❆❧t❤♦✉❣❤ t❤❡ ❞❡t❛✐❧s ❛r❡ tr✐❝❦②✱ ✇❡ ❝❛♥ ❛♣♣❧② ❇❡r♥♦✉❧❧✐ ❜❡t✇❡❡♥ ❍❡❛❞ ▲♦ss❡s t❤❡ ❡♥❞s ♦❢ ❛ ♣✐♣❡ ✇✐t❤ ❱ ❛s t❤❡ ❛✈❡r❛❣❡ ✌♦✇ ✈❡❧♦❝✐t② ✿ ❚✉r❜✉❧❡♥t ❤❡❛❞ ❧♦ss ✔ ♣ ✔ ♣ ✚ ❣ ✰ ❱ ✷ ✚ ❣ ✰ ❱ ✷ ✕ ✕ ❈♦rr❡❧❛t✐♦♥s ✷ ❣ ✰ ❤ ❂ ✷ ❣ ✰ ❤ ✰ ❍ ▲ ❢♦r ❢ ▼✐♥♦r ❧♦ss❡s ❆ ❇ ❚②♣❡s ♦❢ ♣r♦❜❧❡♠ ❍ ▲ ❞❡♣❡♥❞s ✎ ❞✐r❡❝t❧② ♦♥ ▲ ✱ ❞ ✱ ◗ ✷ ✭♦r ❱ ✷ ✮ ✎ ❛❞❞✐t✐♦♥❛❧❧② ♦♥ t❤❡ ❧❡✈❡❧ ♦❢ t✉r❜✉❧❡♥❝❡ ❛♥❞ t❤❡ ♣✐♣❡ ✐♥s✐❞❡ r♦✉❣❤♥❡ss ❚❤✐s ✐s r❡✌❡❝t❡❞ ✐♥ t❤❡ ❉❛r❝②✲❲❡✐s❜❛❝❤ ❡q✉❛t✐♦♥ ❢♦r ❍ ▲ ✓ ❱ ✷ ✒ ✹ ❢ ▲ ❍ ▲ ❂ ❞ ✷ ❣

P✐♣❡ ❋❧♦✇s ■■ ❈♦rr❡❧❛t✐♦♥s ❢♦r ❢ P✐♣❡ ✐♥❧❡t ❍❡❛❞ ▲♦ss❡s ❚✉r❜✉❧❡♥t ❤❡❛❞ ❧♦ss ❢ ✐s t❤❡ ❉❛r❝② ❋r✐❝t✐♦♥ ❋❛❝t♦r ✿ ❢ ❂ ❢ ✭ ✑❀ ❘ ❡ ✮ ❈♦rr❡❧❛t✐♦♥s ❋♦r ❧❛♠✐♥❛r ✌♦✇✱ ✇❡ ❝❛♥ t❛❦❡ ❢♦r ❢ ▼✐♥♦r ❧♦ss❡s ❢ ❂ ✶✻ ❚②♣❡s ♦❢ ❘ ❡ ♣r♦❜❧❡♠ ❋♦r t✉r❜✉❧❡♥t ✌♦✇✱ ✐❢ t❤❡ ♣✐♣❡ ✐s s♠♦♦t❤✱ ❢ ❂ ✵ ✿ ✵✼✾ ❢♦r ❘ ❡ ❁ ✶✵ ✺ ❘ ❡ ✵ ✿ ✷✺ ✭♠♦r❡ ❝♦♠♣❧✐❝❛t❡❞ ❡①♣r❡ss✐♦♥s ❡①t❡♥❞ t❤✐s t♦ ❤✐❣❤❡r ✈❛❧✉❡s✮

P✐♣❡ ❋❧♦✇s ■■ ❋♦r r♦✉❣❤ ♣✐♣❡s✱ ❡①♣r❡ss✐♦♥s ✐♥✈♦❧✈✐♥❣ ✑ ❛r❡ ❤✐❞❡♦✉s❧② ❝♦♠♣❧✐❝❛t❡❞✳ ❍♦✇❡✈❡r t❤❡ r❡s✉❧ts ❤❛✈❡ ❜❡❡♥ ♣❧♦tt❡❞ ❛s t❤❡ P✐♣❡ ✐♥❧❡t ▼♦♦❞② ❞✐❛❣r❛♠ ✳ ❍❡❛❞ ▲♦ss❡s ◆✳❇✳ ✿ ✑ ✐s t❤❡ r❛t✐♦ ♦❢ t❤❡ t②♣✐❝❛❧ r♦✉❣❤♥❡ss ❤❡✐❣❤t ❦ t♦ t❤❡ ❚✉r❜✉❧❡♥t ❞✐❛♠❡t❡r ♦❢ t❤❡ ♣✐♣❡ ❞ ✳ ❤❡❛❞ ❧♦ss ❈♦rr❡❧❛t✐♦♥s Moody diagram ❢♦r ❢ ▼✐♥♦r ❧♦ss❡s 0.02 ❚②♣❡s ♦❢ 0.04 0.015 ♣r♦❜❧❡♠ 0.02 s 0.01 s e 0.01 n Friction factor f h g 0.008 u o R 0.004 e v 0.006 i t a l e R 0.001 0.004 0.0004 0.0001 0.002 3 4 5 6 7 10 10 10 10 10 Re

P✐♣❡ ❋❧♦✇s ■■ ▼✐♥♦r ❧♦ss❡s P✐♣❡ ✐♥❧❡t ❍❡❛❞ ▲♦ss❡s ❙♦ ❢❛r ✇❡ ❤❛✈❡ ✐❣♥♦r❡❞ t❤❡ ❡♥tr❛♥❝❡ s❡❝t✐♦♥✳ ❲❡ ❤❛✈❡ ❛❧s♦ ❚✉r❜✉❧❡♥t ✐❣♥♦r❡❞ ✿ ❤❡❛❞ ❧♦ss ✎ P✐♣❡ ❜❡♥❞s ❈♦rr❡❧❛t✐♦♥s ❢♦r ❢ ✎ ❈❤❛♥❣❡s ✐♥ ❞✐❛♠❡t❡r ✭s✉❞❞❡♥✴❣r❛❞✉❛❧ ▼✐♥♦r ❧♦ss❡s ❡①♣❛♥s✐♦♥s✴❝♦♥tr❛❝t✐♦♥s✮ ❚②♣❡s ♦❢ ♣r♦❜❧❡♠ ✎ ❱❛❧✈❡s ❚❤❡s❡ ❛r❡ r❡❢❡rr❡❞ t♦ ❛s ♠✐♥♦r ❧♦ss❡s ✭❜✉t ❝❛♥ ❜❡ q✉✐t❡ ❧❛r❣❡✮✳ ❋♦r ❡❛❝❤ ❡❧❡♠❡♥t ♦❢ t❤❡ ♣✐♣❡ s②st❡♠✱ ✐♥tr♦❞✉❝❡ ❛ ♠✐♥♦r ❧♦ss t❡r♠ ❤ ❧ ❂ ❑ ❱ ✷ ✷ ❣ ❑ ✐s ❛ ❞✐♠❡♥s✐♦♥❧❡ss ❝♦♥st❛♥t✱ ❞❡t❡r♠✐♥❡❞ ❡①♣❡r✐♠❡♥t❛❧❧② ❛♥❞ t❛❜✉❧❛t❡❞✳